# Results of Zeta and Related Functions

DLMF Formula Maple Mathematica Symbolic
Maple
Symbolic
Mathematica
Numeric
Maple
Numeric
Mathematica
25.2.E1 ${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{n=1}^{\infty}\frac{1}{n^% {s}}}}$ Zeta(s)= sum((1)/((n)^(s)), n = 1..infinity) Zeta[s]= Sum[Divide[1,(n)^(s)], {n, 1, Infinity}] Failure Successful Skip -
25.2.E2 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{-s}}\sum_{n=0}^{% \infty}\frac{1}{(2n+1)^{s}}}}$ Zeta(s)=(1)/(1 - (2)^(- s))*sum((1)/((2*n + 1)^(s)), n = 0..infinity) Zeta[s]=Divide[1,1 - (2)^(- s)]*Sum[Divide[1,(2*n + 1)^(s)], {n, 0, Infinity}] Successful Successful - -
25.2.E3 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{1-2^{1-s}}\sum_{n=1}^% {\infty}\frac{(-1)^{n-1}}{n^{s}}}}$ Zeta(s)=(1)/(1 - (2)^(1 - s))*sum(((- 1)^(n - 1))/((n)^(s)), n = 1..infinity) Zeta[s]=Divide[1,1 - (2)^(1 - s)]*Sum[Divide[(- 1)^(n - 1),(n)^(s)], {n, 1, Infinity}] Failure Successful Skip -
25.2.E4 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\sum_{n=0}^{% \infty}\frac{(-1)^{n}}{n!}\gamma_{n}(s-1)^{n}}}$ Zeta(s)=(1)/(s - 1)+ sum(((- 1)^(n))/(factorial(n))*gamma[n]*(s - 1)^(n), n = 0..infinity) Zeta[s]=Divide[1,s - 1]+ Sum[Divide[(- 1)^(n),(n)!]*Subscript[\[Gamma], n]*(s - 1)^(n), {n, 0, Infinity}] Failure Failure Skip Skip
25.2.E6 ${\displaystyle{\displaystyle\zeta'\left(s\right)=-\sum_{n=2}^{\infty}(\ln n)n^% {-s}}}$ subs( temp=s, diff( Zeta(temp), temp$(1) ) )= - sum((ln(n))* (n)^(- s), n = 2..infinity) (D[Zeta[temp], {temp, 1}]/.temp-> s)= - Sum[(Log[n])* (n)^(- s), {n, 2, Infinity}] Successful Successful - - 25.2.E7 ${\displaystyle{\displaystyle{\zeta^{(k)}}\left(s\right)=(-1)^{k}\sum_{n=2}^{% \infty}(\ln n)^{k}n^{-s}}}$ subs( temp=s, diff( Zeta(temp), temp$(k) ) )=(- 1)^(k)* sum((ln(n))^(k)* (n)^(- s), n = 2..infinity) (D[Zeta[temp], {temp, k}]/.temp-> s)=(- 1)^(k)* Sum[(Log[n])^(k)* (n)^(- s), {n, 2, Infinity}] Failure Failure Skip Successful
25.2.E8 ${\displaystyle{\displaystyle\zeta\left(s\right)=\sum_{k=1}^{N}\frac{1}{k^{s}}+% \frac{N^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right\rfloor}{x^{% s+1}}\mathrm{d}x}}$ Zeta(s)= sum((1)/((k)^(s)), k = 1..N)+((N)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x)^(s + 1)), x = N..infinity) Zeta[s]= Sum[Divide[1,(k)^(s)], {k, 1, N}]+Divide[(N)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x)^(s + 1)], {x, N, Infinity}] Failure Failure Skip Successful
25.2.E11 ${\displaystyle{\displaystyle\zeta\left(s\right)=\prod_{p}(1-p^{-s})^{-1}}}$ Zeta(s)= product((1 - (p)^(- s))^(- 1), p = - infinity..infinity) Zeta[s]= Product[(1 - (p)^(- s))^(- 1), {p, - Infinity, Infinity}] Failure Failure Skip -
25.2.E12 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{(2\pi)^{s}e^{-s-(\gamma s% /2)}}{2(s-1)\Gamma\left(\tfrac{1}{2}s+1\right)}\prod_{\rho}\left(1-\frac{s}{% \rho}\right)e^{s/\rho}}}$ Zeta(s)=((2*Pi)^(s)* exp(- s -(gamma*s/ 2)))/(2*(s - 1)* GAMMA((1)/(2)*s + 1))*product((1 -(s)/(rho))* exp(s/ rho), rho = - infinity..infinity) Zeta[s]=Divide[(2*Pi)^(s)* Exp[- s -(EulerGamma*s/ 2)],2*(s - 1)* Gamma[Divide[1,2]*s + 1]]*Product[(1 -Divide[s,\[Rho]])* Exp[s/ \[Rho]], {\[Rho], - Infinity, Infinity}] Failure Failure Skip Skip
25.4.E1 ${\displaystyle{\displaystyle\zeta\left(1-s\right)=2(2\pi)^{-s}\cos\left(\tfrac% {1}{2}\pi s\right)\Gamma\left(s\right)\zeta\left(s\right)}}$ Zeta(1 - s)= 2*(2*Pi)^(- s)* cos((1)/(2)*Pi*s)*GAMMA(s)*Zeta(s) Zeta[1 - s]= 2*(2*Pi)^(- s)* Cos[Divide[1,2]*Pi*s]*Gamma[s]*Zeta[s] Failure Successful Successful -
25.4.E2 ${\displaystyle{\displaystyle\zeta\left(s\right)=2(2\pi)^{s-1}\sin\left(\tfrac{% 1}{2}\pi s\right)\Gamma\left(1-s\right)\zeta\left(1-s\right)}}$ Zeta(s)= 2*(2*Pi)^(s - 1)* sin((1)/(2)*Pi*s)*GAMMA(1 - s)*Zeta(1 - s) Zeta[s]= 2*(2*Pi)^(s - 1)* Sin[Divide[1,2]*Pi*s]*Gamma[1 - s]*Zeta[1 - s] Failure Successful Successful -
25.4.E3 ${\displaystyle{\displaystyle\xi\left(s\right)=\xi\left(1-s\right)}}$ (s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 = (1 - s)*(1 - s-1)*GAMMA((1 - s)/2)*Pi^(-(1 - s)/2)*Zeta(1 - s)/2 Error Failure Error Successful -
25.4.E4 ${\displaystyle{\displaystyle\xi\left(s\right)=\tfrac{1}{2}s(s-1)\Gamma\left(% \tfrac{1}{2}s\right)\pi^{-s/2}\zeta\left(s\right)}}$ (s)*(s-1)*GAMMA((s)/2)*Pi^(-(s)/2)*Zeta(s)/2 =(1)/(2)*s*(s - 1)* GAMMA((1)/(2)*s)*(Pi)^(- s/ 2)* Zeta(s) Error Successful Error - -
25.4.E5 ${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-s\right)=\frac{2}{(2% \pi)^{s}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}\genfrac{(}{% )}{0.0pt}{}{m}{r}\left(\Re(c^{k-m})\cos\left(\tfrac{1}{2}\pi s\right)+\Im(c^{k% -m})\sin\left(\tfrac{1}{2}\pi s\right)\right){\Gamma^{(r)}}\left(s\right){% \zeta^{(m-r)}}\left(s\right)}}$ (- 1)^(k)* subs( temp=1 - s, diff( Zeta(temp), temp$(k) ) )=(2)/((2*Pi)^(s))*sum(sum(binomial(k,m)*binomial(m,r)*(Re((c)^(k - m))*cos((1)/(2)*Pi*s)+ Im((c)^(k - m))*sin((1)/(2)*Pi*s))* subs( temp=s, diff( GAMMA(temp), temp$(r) ) )*subs( temp=s, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k) (- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - s)=Divide[2,(2*Pi)^(s)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*(Re[(c)^(k - m)]*Cos[Divide[1,2]*Pi*s]+ Im[(c)^(k - m)]*Sin[Divide[1,2]*Pi*s])* (D[Gamma[temp], {temp, r}]/.temp-> s)*(D[Zeta[temp], {temp, m - r}]/.temp-> s), {r, 0, m}], {m, 0, k}] Failure Failure Skip Skip 25.4.E6 ${\displaystyle{\displaystyle c=-\ln\left(2\pi\right)-\tfrac{1}{2}\pi\mathrm{i}}}$ c = - ln(2*Pi)-(1)/(2)*Pi*I c = - Log[2*Pi]-Divide[1,2]*Pi*I Failure Failure Fail 3.252090629+2.985009889*I <- {c = 2^(1/2)+I*2^(1/2)} 3.252090629+.156582765*I <- {c = 2^(1/2)-I*2^(1/2)} .423663505+.156582765*I <- {c = -2^(1/2)-I*2^(1/2)} .423663505+2.985009889*I <- {c = -2^(1/2)+I*2^(1/2)} Fail Complex[3.2520906287824403, 2.9850098891679915] <- {Rule[c, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[3.2520906287824403, 0.1565827644218014] <- {Rule[c, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} Complex[0.4236635040362502, 0.1565827644218014] <- {Rule[c, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]} Complex[0.4236635040362502, 2.9850098891679915] <- {Rule[c, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]} 25.5.E1 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s\right)}% \int_{0}^{\infty}\frac{x^{s-1}}{e^{x}-1}\mathrm{d}x}}$ Zeta(s)=(1)/(GAMMA(s))*int(((x)^(s - 1))/(exp(x)- 1), x = 0..infinity) Zeta[s]=Divide[1,Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]- 1], {x, 0, Infinity}] Failure Successful Skip - 25.5.E2 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{\Gamma\left(s+1\right% )}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}-1)^{2}}\mathrm{d}x}}$ Zeta(s)=(1)/(GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)- 1)^(2)), x = 0..infinity) Zeta[s]=Divide[1,Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]- 1)^(2)], {x, 0, Infinity}] Failure Failure Skip Skip 25.5.E3 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{e^{x}+1}\mathrm{d}x}}$ Zeta(s)=(1)/((1 - (2)^(1 - s))* GAMMA(s))*int(((x)^(s - 1))/(exp(x)+ 1), x = 0..infinity) Zeta[s]=Divide[1,(1 - (2)^(1 - s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Exp[x]+ 1], {x, 0, Infinity}] Failure Successful Skip - 25.5.E4 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{(1-2^{1-s})\Gamma% \left(s+1\right)}\int_{0}^{\infty}\frac{e^{x}x^{s}}{(e^{x}+1)^{2}}\mathrm{d}x}}$ Zeta(s)=(1)/((1 - (2)^(1 - s))* GAMMA(s + 1))*int((exp(x)*(x)^(s))/((exp(x)+ 1)^(2)), x = 0..infinity) Zeta[s]=Divide[1,(1 - (2)^(1 - s))* Gamma[s + 1]]*Integrate[Divide[Exp[x]*(x)^(s),(Exp[x]+ 1)^(2)], {x, 0, Infinity}] Failure Failure Skip Skip 25.5.E5 ${\displaystyle{\displaystyle\zeta\left(s\right)=-s\int_{0}^{\infty}\frac{x-% \left\lfloor x\right\rfloor-\frac{1}{2}}{x^{s+1}}\mathrm{d}x}}$ Zeta(s)= - s*int((x - floor(x)-(1)/(2))/((x)^(s + 1)), x = 0..infinity) Zeta[s]= - s*Integrate[Divide[x - Floor[x]-Divide[1,2],(x)^(s + 1)], {x, 0, Infinity}] Failure Failure Skip Successful 25.5.E6 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+% \frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1% }{x}+\frac{1}{2}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}$ Zeta(s)=(1)/(2)+(1)/(s - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2))*((x)^(s - 1))/(exp(x)), x = 0..infinity) Zeta[s]=Divide[1,2]+Divide[1,s - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}] Failure Failure Skip Successful 25.5.E7 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+\sum% _{m=1}^{n}\frac{B_{2m}}{(2m)!}{\left(s\right)_{2m-1}}+\frac{1}{\Gamma\left(s% \right)}\int_{0}^{\infty}\left(\frac{1}{e^{x}-1}-\frac{1}{x}+\frac{1}{2}-\sum_% {m=1}^{n}\frac{B_{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{e^{x}}\mathrm{d}x}}$ Zeta(s)=(1)/(2)+(1)/(s - 1)+ sum((bernoulli(2*m))/(factorial(2*m))*pochhammer(s, 2*m - 1)+(1)/(GAMMA(s))*int(((1)/(exp(x)- 1)-(1)/(x)+(1)/(2)- sum((bernoulli(2*m))/(factorial(2*m))*(x)^(2*m - 1), m = 1..n))*((x)^(s - 1))/(exp(x)), x = 0..infinity), m = 1..n) Zeta[s]=Divide[1,2]+Divide[1,s - 1]+ Sum[Divide[BernoulliB[2*m],(2*m)!]*Pochhammer[s, 2*m - 1]+Divide[1,Gamma[s]]*Integrate[(Divide[1,Exp[x]- 1]-Divide[1,x]+Divide[1,2]- Sum[Divide[BernoulliB[2*m],(2*m)!]*(x)^(2*m - 1), {m, 1, n}])*Divide[(x)^(s - 1),Exp[x]], {x, 0, Infinity}], {m, 1, n}] Failure Failure Skip Error 25.5.E8 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2(1-2^{-s})\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh x}\mathrm{d}x}}$ Zeta(s)=(1)/(2*(1 - (2)^(- s))* GAMMA(s))*int(((x)^(s - 1))/(sinh(x)), x = 0..infinity) Zeta[s]=Divide[1,2*(1 - (2)^(- s))* Gamma[s]]*Integrate[Divide[(x)^(s - 1),Sinh[x]], {x, 0, Infinity}] Failure Failure Skip Skip 25.5.E9 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{\Gamma\left(s+1% \right)}\int_{0}^{\infty}\frac{x^{s}}{(\sinh x)^{2}}\mathrm{d}x}}$ Zeta(s)=((2)^(s - 1))/(GAMMA(s + 1))*int(((x)^(s))/((sinh(x))^(2)), x = 0..infinity) Zeta[s]=Divide[(2)^(s - 1),Gamma[s + 1]]*Integrate[Divide[(x)^(s),(Sinh[x])^(2)], {x, 0, Infinity}] Failure Failure Skip - 25.5.E10 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{1-2^{1-s}}\int_% {0}^{\infty}\frac{\cos\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}% \cosh\left(\frac{1}{2}\pi x\right)}\mathrm{d}x}}$ Zeta(s)=((2)^(s - 1))/(1 - (2)^(1 - s))*int((cos(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)* cosh((1)/(2)*Pi*x)), x = 0..infinity) Zeta[s]=Divide[(2)^(s - 1),1 - (2)^(1 - s)]*Integrate[Divide[Cos[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)* Cosh[Divide[1,2]*Pi*x]], {x, 0, Infinity}] Failure Failure Skip Error 25.5.E11 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{2}+\frac{1}{s-1}+2% \int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/% 2}(e^{2\pi x}-1)}\mathrm{d}x}}$ Zeta(s)=(1)/(2)+(1)/(s - 1)+ 2*int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(2*Pi*x)- 1)), x = 0..infinity) Zeta[s]=Divide[1,2]+Divide[1,s - 1]+ 2*Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[2*Pi*x]- 1)], {x, 0, Infinity}] Failure Successful Skip - 25.5.E12 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{2^{s-1}}{s-1}-2^{s}\int_% {0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2})^{s/2}(e^% {\pi x}+1)}\mathrm{d}x}}$ Zeta(s)=((2)^(s - 1))/(s - 1)- (2)^(s)* int((sin(s*arctan(x)))/((1 + (x)^(2))^(s/ 2)*(exp(Pi*x)+ 1)), x = 0..infinity) Zeta[s]=Divide[(2)^(s - 1),s - 1]- (2)^(s)* Integrate[Divide[Sin[s*ArcTan[x]],(1 + (x)^(2))^(s/ 2)*(Exp[Pi*x]+ 1)], {x, 0, Infinity}] Failure Successful Skip - 25.5.E13 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\pi^{s/2}}{s(s-1)\Gamma% \left(\frac{1}{2}s\right)}+\frac{\pi^{s/2}}{\Gamma\left(\frac{1}{2}s\right)}\*% \int_{1}^{\infty}\left(x^{s/2}+x^{(1-s)/2}\right)\frac{\omega(x)}{x}\mathrm{d}% x}}$ Zeta(s)=((Pi)^(s/ 2))/(s*(s - 1)* GAMMA((1)/(2)*s))+((Pi)^(s/ 2))/(GAMMA((1)/(2)*s))* int(((x)^(s/ 2)+ (x)^((1 - s)/ 2))*(omega*(x))/(x), x = 1..infinity) Zeta[s]=Divide[(Pi)^(s/ 2),s*(s - 1)* Gamma[Divide[1,2]*s]]+Divide[(Pi)^(s/ 2),Gamma[Divide[1,2]*s]]* Integrate[((x)^(s/ 2)+ (x)^((1 - s)/ 2))*Divide[\[Omega]*(x),x], {x, 1, Infinity}] Failure Failure Skip Skip 25.5.E14 ${\displaystyle{\displaystyle\omega(x)=\sum_{n=1}^{\infty}e^{-n^{2}\pi x}}}$ omega*(x)= sum(exp(- (n)^(2)* Pi*x), n = 1..infinity) \[Omega]*(x)= Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}] Failure Failure Skip Fail Complex[1.370996156766441, 1.4142135623730951] <- {Rule[x, 1], Rule[Ο, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[2.826559682002321, 2.8284271247461903] <- {Rule[x, 2], Rule[Ο, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[4.242559987601715, 4.242640687119286] <- {Rule[x, 3], Rule[Ο, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]} Complex[1.370996156766441, -1.4142135623730951] <- {Rule[x, 1], Rule[Ο, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]} ... skip entries to safe data 25.5.E14 ${\displaystyle{\displaystyle\sum_{n=1}^{\infty}e^{-n^{2}\pi x}=\frac{1}{2}% \left(\theta_{3}\left(0\middle|ix\right)-1\right)}}$ sum(exp(- (n)^(2)* Pi*x), n = 1..infinity)=(1)/(2)*(JacobiTheta3(0,exp(I*Pi*I*x))- 1) Sum[Exp[- (n)^(2)* Pi*x], {n, 1, Infinity}]=Divide[1,2]*(EllipticTheta[3, 0, I*x]- 1) Failure Failure Skip Successful 25.5.E15 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(% \pi s\right)}{\pi}\*\int_{0}^{\infty}(\ln\left(1+x\right)-\psi\left(1+x\right)% )x^{-s}\mathrm{d}x}}$ Zeta(s)=(1)/(s - 1)+(sin(Pi*s))/(Pi)* int((ln(1 + x)- Psi(1 + x))* (x)^(- s), x = 0..infinity) Zeta[s]=Divide[1,s - 1]+Divide[Sin[Pi*s],Pi]* Integrate[(Log[1 + x]- PolyGamma[1 + x])* (x)^(- s), {x, 0, Infinity}] Failure Failure Skip Error 25.5.E16 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{1}{s-1}+\frac{\sin\left(% \pi s\right)}{\pi(s-1)}\*\int_{0}^{\infty}\left(\frac{1}{1+x}-\psi'\left(1+x% \right)\right)x^{1-s}\mathrm{d}x}}$ Zeta(s)=(1)/(s - 1)+(sin(Pi*s))/(Pi*(s - 1))* int(((1)/(1 + x)- subs( temp=1 + x, diff( Psi(temp), temp$(1) ) ))* (x)^(1 - s), x = 0..infinity) Zeta[s]=Divide[1,s - 1]+Divide[Sin[Pi*s],Pi*(s - 1)]* Integrate[(Divide[1,1 + x]- (D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x))* (x)^(1 - s), {x, 0, Infinity}] Failure Failure Skip -
25.5.E17 ${\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)% }{\pi}\int_{0}^{\infty}\left(\gamma+\psi\left(1+x\right)\right)x^{-s-1}\mathrm% {d}x}}$ Zeta(1 + s)=(sin(Pi*s))/(Pi)*int((gamma + Psi(1 + x))* (x)^(- s - 1), x = 0..infinity) Zeta[1 + s]=Divide[Sin[Pi*s],Pi]*Integrate[(EulerGamma + PolyGamma[1 + x])* (x)^(- s - 1), {x, 0, Infinity}] Failure Failure Skip Error
25.5.E18 ${\displaystyle{\displaystyle\zeta\left(1+s\right)=\frac{\sin\left(\pi s\right)% }{\pi s}\int_{0}^{\infty}\psi'\left(1+x\right)x^{-s}\mathrm{d}x}}$ Zeta(1 + s)=(sin(Pi*s))/(Pi*s)*int(subs( temp=1 + x, diff( Psi(temp), temp$(1) ) )*(x)^(- s), x = 0..infinity) Zeta[1 + s]=Divide[Sin[Pi*s],Pi*s]*Integrate[(D[PolyGamma[temp], {temp, 1}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}] Failure Failure Skip Skip 25.5.E19 ${\displaystyle{\displaystyle\zeta\left(m+s\right)=(-1)^{m-1}\frac{\Gamma\left(% s\right)\sin\left(\pi s\right)}{\pi\Gamma\left(m+s\right)}\*\int_{0}^{\infty}{% \psi^{(m)}}\left(1+x\right)x^{-s}\mathrm{d}x}}$ Zeta(m + s)=(- 1)^(m - 1)*(GAMMA(s)*sin(Pi*s))/(Pi*GAMMA(m + s))* int(subs( temp=1 + x, diff( Psi(temp), temp$(m) ) )*(x)^(- s), x = 0..infinity) Zeta[m + s]=(- 1)^(m - 1)*Divide[Gamma[s]*Sin[Pi*s],Pi*Gamma[m + s]]* Integrate[(D[PolyGamma[temp], {temp, m}]/.temp-> 1 + x)*(x)^(- s), {x, 0, Infinity}] Failure Failure Skip Error
25.5.E20 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{% 2\pi i}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}-1}\mathrm{d}z}}$ Zeta(s)=(GAMMA(1 - s))/(2*Pi*I)*int(((z)^(s - 1))/(exp(- z)- 1), z = - infinity..(0 +)) Zeta[s]=Divide[Gamma[1 - s],2*Pi*I]*Integrate[Divide[(z)^(s - 1),Exp[- z]- 1], {z, - Infinity, (0 +)}] Error Failure - Error
25.5.E21 ${\displaystyle{\displaystyle\zeta\left(s\right)=\frac{\Gamma\left(1-s\right)}{% 2\pi i(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{e^{-z}+1}\mathrm{d}z}}$ Zeta(s)=(GAMMA(1 - s))/(2*Pi*I*(1 - (2)^(1 - s)))* int(((z)^(s - 1))/(exp(- z)+ 1), z = - infinity..(0 +)) Zeta[s]=Divide[Gamma[1 - s],2*Pi*I*(1 - (2)^(1 - s))]* Integrate[Divide[(z)^(s - 1),Exp[- z]+ 1], {z, - Infinity, (0 +)}] Error Failure - Error
25.6#Ex1 ${\displaystyle{\displaystyle\zeta\left(0\right)=-\frac{1}{2}}}$ Zeta(0)= -(1)/(2) Zeta[0]= -Divide[1,2] Successful Successful - -
25.6#Ex2 ${\displaystyle{\displaystyle\zeta\left(2\right)=\frac{\pi^{2}}{6}}}$ Zeta(2)=((Pi)^(2))/(6) Zeta[2]=Divide[(Pi)^(2),6] Successful Successful - -
25.6#Ex3 ${\displaystyle{\displaystyle\zeta\left(4\right)=\frac{\pi^{4}}{90}}}$ Zeta(4)=((Pi)^(4))/(90) Zeta[4]=Divide[(Pi)^(4),90] Successful Successful - -
25.6#Ex4 ${\displaystyle{\displaystyle\zeta\left(6\right)=\frac{\pi^{6}}{945}}}$ Zeta(6)=((Pi)^(6))/(945) Zeta[6]=Divide[(Pi)^(6),945] Successful Successful - -
25.6.E2 ${\displaystyle{\displaystyle\zeta\left(2n\right)=\frac{(2\pi)^{2n}}{2(2n)!}% \left|B_{2n}\right|}}$ Zeta(2*n)=((2*Pi)^(2*n))/(2*factorial(2*n))*abs(bernoulli(2*n)) Zeta[2*n]=Divide[(2*Pi)^(2*n),2*(2*n)!]*Abs[BernoulliB[2*n]] Failure Failure Successful Successful
25.6.E3 ${\displaystyle{\displaystyle\zeta\left(-n\right)=-\frac{B_{n+1}}{n+1}}}$ Zeta(- n)= -(bernoulli(n + 1))/(n + 1) Zeta[- n]= -Divide[BernoulliB[n + 1],n + 1] Failure Failure Successful Successful
25.6.E4 ${\displaystyle{\displaystyle\zeta\left(-2n\right)=0}}$ Zeta(- 2*n)= 0 Zeta[- 2*n]= 0 Failure Failure Successful Successful
25.6.E6 ${\displaystyle{\displaystyle\zeta\left(2k+1\right)=\frac{(-1)^{k+1}(2\pi)^{2k+% 1}}{2(2k+1)!}\int_{0}^{1}B_{2k+1}\left(t\right)\cot\left(\pi t\right)\mathrm{d% }t}}$ Zeta(2*k + 1)=((- 1)^(k + 1)*(2*Pi)^(2*k + 1))/(2*factorial(2*k + 1))*int(bernoulli(2*k + 1, t)*cot(Pi*t), t = 0..1) Zeta[2*k + 1]=Divide[(- 1)^(k + 1)*(2*Pi)^(2*k + 1),2*(2*k + 1)!]*Integrate[BernoulliB[2*k + 1, t]*Cot[Pi*t], {t, 0, 1}] Failure Failure Skip Successful
25.6.E7 ${\displaystyle{\displaystyle\zeta\left(2\right)=\int_{0}^{1}\int_{0}^{1}\frac{% 1}{1-xy}\mathrm{d}x\mathrm{d}y}}$ Zeta(2)= int(int((1)/(1 - x*y), x = 0..1), y = 0..1) Zeta[2]= Integrate[Integrate[Divide[1,1 - x*y], {x, 0, 1}], {y, 0, 1}] Successful Successful - -
25.6.E8 ${\displaystyle{\displaystyle\zeta\left(2\right)=3\sum_{k=1}^{\infty}\frac{1}{k% ^{2}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$ Zeta(2)= 3*sum((1)/((k)^(2)*binomial(2*k,k)), k = 1..infinity) Zeta[2]= 3*Sum[Divide[1,(k)^(2)*Binomial[2*k,k]], {k, 1, Infinity}] Successful Successful - -
25.6.E9 ${\displaystyle{\displaystyle\zeta\left(3\right)=\frac{5}{2}\sum_{k=1}^{\infty}% \frac{(-1)^{k-1}}{k^{3}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$ Zeta(3)=(5)/(2)*sum(((- 1)^(k - 1))/((k)^(3)*binomial(2*k,k)), k = 1..infinity) Zeta[3]=Divide[5,2]*Sum[Divide[(- 1)^(k - 1),(k)^(3)*Binomial[2*k,k]], {k, 1, Infinity}] Failure Successful Skip -
25.6.E10 ${\displaystyle{\displaystyle\zeta\left(4\right)=\frac{36}{17}\sum_{k=1}^{% \infty}\frac{1}{k^{4}\genfrac{(}{)}{0.0pt}{}{2k}{k}}}}$ Zeta(4)=(36)/(17)*sum((1)/((k)^(4)*binomial(2*k,k)), k = 1..infinity) Zeta[4]=Divide[36,17]*Sum[Divide[1,(k)^(4)*Binomial[2*k,k]], {k, 1, Infinity}] Failure Successful Skip -
25.6.E11 ${\displaystyle{\displaystyle\zeta'\left(0\right)=-\tfrac{1}{2}\ln\left(2\pi% \right)}}$ subs( temp=0, diff( Zeta(temp), temp$(1) ) )= -(1)/(2)*ln(2*Pi) (D[Zeta[temp], {temp, 1}]/.temp-> 0)= -Divide[1,2]*Log[2*Pi] Successful Successful - - 25.6.E12 ${\displaystyle{\displaystyle\zeta''\left(0\right)=-\tfrac{1}{2}(\ln\left(2\pi% \right))^{2}+\tfrac{1}{2}{\gamma^{2}}-\tfrac{1}{24}\pi^{2}+\gamma_{1}}}$ subs( temp=0, diff( Zeta(temp), temp$(2) ) )= -(1)/(2)*(ln(2*Pi))^(2)+(1)/(2)*(gamma)^(2)-(1)/(24)*(Pi)^(2)+ gamma[1] (D[Zeta[temp], {temp, 2}]/.temp-> 0)= -Divide[1,2]*(Log[2*Pi])^(2)+Divide[1,2]*(EulerGamma)^(2)-Divide[1,24]*(Pi)^(2)+ Subscript[\[Gamma], 1] Failure Failure
Fail
-1.487029407-1.414213562*I <- {gamma[1] = 2^(1/2)+I*2^(1/2)}
-1.487029407+1.414213562*I <- {gamma[1] = 2^(1/2)-I*2^(1/2)}
1.341397717+1.414213562*I <- {gamma[1] = -2^(1/2)-I*2^(1/2)}
1.341397717-1.414213562*I <- {gamma[1] = -2^(1/2)+I*2^(1/2)}
Fail
Complex[-1.487029407856772, -1.4142135623730951] <- {Rule[Subscript[Ξ³, 1], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[-1.487029407856772, 1.4142135623730951] <- {Rule[Subscript[Ξ³, 1], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.3413977168894184, 1.4142135623730951] <- {Rule[Subscript[Ξ³, 1], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[1.3413977168894184, -1.4142135623730951] <- {Rule[Subscript[Ξ³, 1], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
25.6.E13 ${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(-2n\right)=\frac{2(-1)^% {n}}{(2\pi)^{2n+1}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}% \genfrac{(}{)}{0.0pt}{}{m}{r}\Im(c^{k-m})\*{\Gamma^{(r)}}\left(2n+1\right){% \zeta^{(m-r)}}\left(2n+1\right)}}$ (- 1)^(k)* subs( temp=- 2*n, diff( Zeta(temp), temp$(k) ) )=(2*(- 1)^(n))/((2*Pi)^(2*n + 1))*sum(sum(binomial(k,m)*binomial(m,r)*Im((c)^(k - m))* subs( temp=2*n + 1, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n + 1, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k) (- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> - 2*n)=Divide[2*(- 1)^(n),(2*Pi)^(2*n + 1)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Im[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n + 1)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n + 1), {r, 0, m}], {m, 0, k}] Failure Failure Skip Skip 25.6.E14 ${\displaystyle{\displaystyle(-1)^{k}{\zeta^{(k)}}\left(1-2n\right)=\frac{2(-1)% ^{n}}{(2\pi)^{2n}}\sum_{m=0}^{k}\sum_{r=0}^{m}\genfrac{(}{)}{0.0pt}{}{k}{m}% \genfrac{(}{)}{0.0pt}{}{m}{r}\Re(c^{k-m})\*{\Gamma^{(r)}}\left(2n\right){\zeta% ^{(m-r)}}\left(2n\right)}}$ (- 1)^(k)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(k) ) )=(2*(- 1)^(n))/((2*Pi)^(2*n))*sum(sum(binomial(k,m)*binomial(m,r)*Re((c)^(k - m))* subs( temp=2*n, diff( GAMMA(temp), temp$(r) ) )*subs( temp=2*n, diff( Zeta(temp), temp$(m - r) ) ), r = 0..m), m = 0..k) (- 1)^(k)* (D[Zeta[temp], {temp, k}]/.temp-> 1 - 2*n)=Divide[2*(- 1)^(n),(2*Pi)^(2*n)]*Sum[Sum[Binomial[k,m]*Binomial[m,r]*Re[(c)^(k - m)]* (D[Gamma[temp], {temp, r}]/.temp-> 2*n)*(D[Zeta[temp], {temp, m - r}]/.temp-> 2*n), {r, 0, m}], {m, 0, k}] Failure Failure Skip Skip
25.6.E15 ${\displaystyle{\displaystyle\zeta'\left(2n\right)=\frac{(-1)^{n+1}(2\pi)^{2n}}% {2(2n)!}\left(2n\zeta'\left(1-2n\right)-(\psi\left(2n\right)-\ln\left(2\pi% \right))B_{2n}\right)}}$ subs( temp=2*n, diff( Zeta(temp), temp$(1) ) )=((- 1)^(n + 1)*(2*Pi)^(2*n))/(2*factorial(2*n))*(2*n*subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) )-(Psi(2*n)- ln(2*Pi))*bernoulli(2*n)) (D[Zeta[temp], {temp, 1}]/.temp-> 2*n)=Divide[(- 1)^(n + 1)*(2*Pi)^(2*n),2*(2*n)!]*(2*n*(D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n)-(PolyGamma[2*n]- Log[2*Pi])*BernoulliB[2*n]) Failure Failure Successful Successful
25.6.E16 ${\displaystyle{\displaystyle\left(n+\tfrac{1}{2}\right)\zeta\left(2n\right)=% \sum_{k=1}^{n-1}\zeta\left(2k\right)\zeta\left(2n-2k\right)}}$ (n +(1)/(2))* Zeta(2*n)= sum(Zeta(2*k)*Zeta(2*n - 2*k), k = 1..n - 1) (n +Divide[1,2])* Zeta[2*n]= Sum[Zeta[2*k]*Zeta[2*n - 2*k], {k, 1, n - 1}] Failure Failure Skip Successful
25.6.E17 ${\displaystyle{\displaystyle\left(n+\tfrac{3}{4}\right)\zeta\left(4n+2\right)=% \sum_{k=1}^{n}\zeta\left(2k\right)\zeta\left(4n+2-2k\right)}}$ (n +(3)/(4))* Zeta(4*n + 2)= sum(Zeta(2*k)*Zeta(4*n + 2 - 2*k), k = 1..n) (n +Divide[3,4])* Zeta[4*n + 2]= Sum[Zeta[2*k]*Zeta[4*n + 2 - 2*k], {k, 1, n}] Failure Failure Skip Successful
25.6.E20 ${\displaystyle{\displaystyle\tfrac{1}{2}(2^{2n}-1)\zeta\left(2n\right)=\sum_{k% =1}^{n-1}(2^{2n-2k}-1)\zeta\left(2n-2k\right)\zeta\left(2k\right)}}$ (1)/(2)*((2)^(2*n)- 1)* Zeta(2*n)= sum(((2)^(2*n - 2*k)- 1)* Zeta(2*n - 2*k)*Zeta(2*k), k = 1..n - 1) Divide[1,2]*((2)^(2*n)- 1)* Zeta[2*n]= Sum[((2)^(2*n - 2*k)- 1)* Zeta[2*n - 2*k]*Zeta[2*k], {k, 1, n - 1}] Failure Failure Skip Successful
25.8.E1 ${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\left(\zeta\left(k\right)-1% \right)=1}}$ sum(Zeta(k)- 1, k = 2..infinity)= 1 Sum[Zeta[k]- 1, {k, 2, Infinity}]= 1 Failure Successful Skip -
25.8.E2 ${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{\Gamma\left(s+k\right)}{(% k+1)!}\left(\zeta\left(s+k\right)-1\right)=\Gamma\left(s-1\right)}}$ sum((GAMMA(s + k))/(factorial(k + 1))*(Zeta(s + k)- 1), k = 0..infinity)= GAMMA(s - 1) Sum[Divide[Gamma[s + k],(k + 1)!]*(Zeta[s + k]- 1), {k, 0, Infinity}]= Gamma[s - 1] Failure Failure Skip Error
25.8.E3 ${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\frac{{\left(s\right)_{k}}\zeta% \left(s+k\right)}{k!2^{s+k}}=(1-2^{-s})\zeta\left(s\right)}}$ sum((pochhammer(s, k)*Zeta(s + k))/(factorial(k)*(2)^(s + k)), k = 0..infinity)=(1 - (2)^(- s))* Zeta(s) Sum[Divide[Pochhammer[s, k]*Zeta[s + k],(k)!*(2)^(s + k)], {k, 0, Infinity}]=(1 - (2)^(- s))* Zeta[s] Failure Failure Skip Successful
25.8.E4 ${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{(-1)^{k}}{k}(\zeta\left(% nk\right)-1)=\ln\left(\prod_{j=0}^{n-1}\Gamma\left(2-e^{(2j+1)\pi i/n}\right)% \right)}}$ sum(((- 1)^(k))/(k)*(Zeta(n*k)- 1), k = 1..infinity)= ln(product(GAMMA(2 - exp((2*j + 1)* Pi*I/ n)), j = 0..n - 1)) Sum[Divide[(- 1)^(k),k]*(Zeta[n*k]- 1), {k, 1, Infinity}]= Log[Product[Gamma[2 - Exp[(2*j + 1)* Pi*I/ n]], {j, 0, n - 1}]] Failure Failure Skip
Fail
Complex[0.7210663818131499, 0.6288153989756469] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.7210663818131499, -2.199611725770543] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.1073607429330403, -2.199611725770543] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-2.1073607429330403, 0.6288153989756469] <- {Rule[Product[Gamma[Plus[2, Times[-1, Power[E, Times[Complex[0, 1], Plus[1, Times[2, j]], Power[n, -1], Pi]]]]], {j, 0, Plus[-1, n]}], Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Sum[Times[Power[-1, k], Power[k, -1], Plus[-1, Zeta[Times[k, n]]]], {k, 1, DirectedInfinity[1]}], Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
25.8.E5 ${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\zeta\left(k\right)z^{k}=-% \gamma z-z\psi\left(1-z\right)}}$ sum(Zeta(k)*(z)^(k), k = 2..infinity)= - gamma*z - z*Psi(1 - z) Sum[Zeta[k]*(z)^(k), {k, 2, Infinity}]= - EulerGamma*z - z*PolyGamma[1 - z] Failure Successful Skip -
25.8.E6 ${\displaystyle{\displaystyle\sum_{k=0}^{\infty}\zeta\left(2k\right)z^{2k}=-% \tfrac{1}{2}\pi z\cot\left(\pi z\right)}}$ sum(Zeta(2*k)*(z)^(2*k), k = 0..infinity)= -(1)/(2)*Pi*z*cot(Pi*z) Sum[Zeta[2*k]*(z)^(2*k), {k, 0, Infinity}]= -Divide[1,2]*Pi*z*Cot[Pi*z] Failure Failure Skip Skip
25.8.E7 ${\displaystyle{\displaystyle\sum_{k=2}^{\infty}\frac{\zeta\left(k\right)}{k}z^% {k}=-\gamma z+\ln\Gamma\left(1-z\right)}}$ sum((Zeta(k))/(k)*(z)^(k), k = 2..infinity)= - gamma*z + ln(GAMMA(1 - z)) Sum[Divide[Zeta[k],k]*(z)^(k), {k, 2, Infinity}]= - EulerGamma*z + Log[Gamma[1 - z]] Failure Successful Skip -
25.8.E8 ${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{k}z% ^{2k}=\ln\left(\frac{\pi z}{\sin\left(\pi z\right)}\right)}}$ sum((Zeta(2*k))/(k)*(z)^(2*k), k = 1..infinity)= ln((Pi*z)/(sin(Pi*z))) Sum[Divide[Zeta[2*k],k]*(z)^(2*k), {k, 1, Infinity}]= Log[Divide[Pi*z,Sin[Pi*z]]] Failure Successful Skip -
25.8.E9 ${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k% +1)2^{2k}}=\frac{1}{2}-\frac{1}{2}\ln 2}}$ sum((Zeta(2*k))/((2*k + 1)* (2)^(2*k)), k = 1..infinity)=(1)/(2)-(1)/(2)*ln(2) Sum[Divide[Zeta[2*k],(2*k + 1)* (2)^(2*k)], {k, 1, Infinity}]=Divide[1,2]-Divide[1,2]*Log[2] Failure Successful Skip -
25.8.E10 ${\displaystyle{\displaystyle\sum_{k=1}^{\infty}\frac{\zeta\left(2k\right)}{(2k% +1)(2k+2)2^{2k}}=\frac{1}{4}-\frac{7}{4\pi^{2}}\zeta\left(3\right)}}$ sum((Zeta(2*k))/((2*k + 1)*(2*k + 2)* (2)^(2*k)), k = 1..infinity)=(1)/(4)-(7)/(4*(Pi)^(2))*Zeta(3) Sum[Divide[Zeta[2*k],(2*k + 1)*(2*k + 2)* (2)^(2*k)], {k, 1, Infinity}]=Divide[1,4]-Divide[7,4*(Pi)^(2)]*Zeta[3] Failure Successful Skip -
25.9.E2 ${\displaystyle{\displaystyle\chi(s)=\pi^{s-\frac{1}{2}}\Gamma\left(\tfrac{1}{2% }-\tfrac{1}{2}s\right)/\Gamma\left(\tfrac{1}{2}s\right)}}$ chi*(s)= (Pi)^(s -(1)/(2))* GAMMA((1)/(2)-(1)/(2)*s)/ GAMMA((1)/(2)*s) \[Chi]*(s)= (Pi)^(s -Divide[1,2])* Gamma[Divide[1,2]-Divide[1,2]*s]/ Gamma[Divide[1,2]*s] Failure Failure
Fail
.5066144201+7.721862512*I <- {chi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)+I*2^(1/2)}
4.506614418-3.721862514*I <- {chi = 2^(1/2)+I*2^(1/2), s = 2^(1/2)-I*2^(1/2)}
-.5006270982e-1-4.069033292*I <- {chi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)-I*2^(1/2)}
-4.050062708+.6903329420e-1*I <- {chi = 2^(1/2)+I*2^(1/2), s = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[0.5066144187413095, 7.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[4.506614418741309, 3.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[0.5066144187413095, -0.2781374851895251] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-3.4933855812586905, 3.721862514810475] <- {Rule[s, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
25.10.E1 ${\displaystyle{\displaystyle Z(t)=\exp\left(i\vartheta(t)\right)\zeta\left(% \tfrac{1}{2}+it\right)}}$ Z*(t)= exp(I*vartheta*(t))*Zeta((1)/(2)+ I*t) Z*(t)= Exp[I*\[CurlyTheta]*(t)]*Zeta[Divide[1,2]+ I*t] Failure Failure
Fail
-.1598353599e-2+4.002319388*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)+I*2^(1/2)}
.1528788606+3.983270213*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = 2^(1/2)-I*2^(1/2)}
-4.764624907+10.91400505*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)-I*2^(1/2)}
-.3879562929e-1+3.851182221*I <- {Z = 2^(1/2)+I*2^(1/2), t = 2^(1/2)+I*2^(1/2), vartheta = -2^(1/2)+I*2^(1/2)}
... skip entries to safe data
Fail
Complex[-0.0015983535965552907, 4.002319390307897] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[1, 1], Power[2, Rational[1, 2]]]]}
Complex[0.15287886062247902, 3.9832702156526483] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[1, -1], Power[2, Rational[1, 2]]]]}
Complex[-4.764624919768366, 10.914005063393518] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[-1, -1], Power[2, Rational[1, 2]]]]}
Complex[-0.03879562949747604, 3.8511822226969143] <- {Rule[t, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Z, Times[Complex[1, 1], Power[2, Rational[1, 2]]]], Rule[Ο, Times[Complex[-1, 1], Power[2, Rational[1, 2]]]]}
... skip entries to safe data
25.10.E3 ${\displaystyle{\displaystyle Z(t)=2\sum_{n=1}^{m}\frac{\cos\left(\vartheta(t)-% t\ln n\right)}{n^{1/2}}+R(t)}}$ Z*(t)= 2*sum((cos(vartheta*(t)- t*ln(n)))/((n)^(1/ 2)), n = 1..m)+ R*(t) Z*(t)= 2*Sum[Divide[Cos[\[CurlyTheta]*(t)- t*Log[n]],(n)^(1/ 2)], {n, 1, m}]+ R*(t) Failure Failure Skip Skip
25.11.E1 ${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{1}{% (n+a)^{s}}}}$ Zeta(0, s, a)= sum((1)/((n + a)^(s)), n = 0..infinity) HurwitzZeta[s, a]= Sum[Divide[1,(n + a)^(s)], {n, 0, Infinity}] Failure Successful Skip -
25.11.E2 ${\displaystyle{\displaystyle\zeta\left(s,1\right)=\zeta\left(s\right)}}$ Zeta(0, s, 1)= Zeta(s) HurwitzZeta[s, 1]= Zeta[s] Successful Successful - -
25.11.E3 ${\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+1\right)+a^{-% s}}}$ Zeta(0, s, a)= Zeta(0, s, a + 1)+ (a)^(- s) HurwitzZeta[s, a]= HurwitzZeta[s, a + 1]+ (a)^(- s) Failure Successful Successful -
25.11.E4 ${\displaystyle{\displaystyle\zeta\left(s,a\right)=\zeta\left(s,a+m\right)+\sum% _{n=0}^{m-1}\frac{1}{(n+a)^{s}}}}$ Zeta(0, s, a)= Zeta(0, s, a + m)+ sum((1)/((n + a)^(s)), n = 0..m - 1) HurwitzZeta[s, a]= HurwitzZeta[s, a + m]+ Sum[Divide[1,(n + a)^(s)], {n, 0, m - 1}] Failure Successful Skip -
25.11.E5 ${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{N}\frac{1}{(n+a)% ^{s}}+\frac{(N+a)^{1-s}}{s-1}-s\int_{N}^{\infty}\frac{x-\left\lfloor x\right% \rfloor}{(x+a)^{s+1}}\mathrm{d}x}}$ Zeta(0, s, a)= sum((1)/((n + a)^(s)), n = 0..N)+((N + a)^(1 - s))/(s - 1)- s*int((x - floor(x))/((x + a)^(s + 1)), x = N..infinity) HurwitzZeta[s, a]= Sum[Divide[1,(n + a)^(s)], {n, 0, N}]+Divide[(N + a)^(1 - s),s - 1]- s*Integrate[Divide[x - Floor[x],(x + a)^(s + 1)], {x, N, Infinity}] Failure Failure Skip Error
25.11.E8 ${\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}a\right)=\zeta\left(s,% \tfrac{1}{2}a+\tfrac{1}{2}\right)+2^{s}\sum_{n=0}^{\infty}\frac{(-1)^{n}}{(n+a% )^{s}}}}$ Zeta(0, s, (1)/(2)*a)= Zeta(0, s, (1)/(2)*a +(1)/(2))+ (2)^(s)* sum(((- 1)^(n))/((n + a)^(s)), n = 0..infinity) HurwitzZeta[s, Divide[1,2]*a]= HurwitzZeta[s, Divide[1,2]*a +Divide[1,2]]+ (2)^(s)* Sum[Divide[(- 1)^(n),(n + a)^(s)], {n, 0, Infinity}] Failure Failure Skip Successful
25.11.E9 ${\displaystyle{\displaystyle\zeta\left(1-s,a\right)=\frac{2\Gamma\left(s\right% )}{(2\pi)^{s}}\*\sum_{n=1}^{\infty}\frac{1}{n^{s}}\cos\left(\tfrac{1}{2}\pi s-% 2n\pi a\right)}}$ Zeta(0, 1 - s, a)=(2*GAMMA(s))/((2*Pi)^(s))* sum((1)/((n)^(s))*cos((1)/(2)*Pi*s - 2*n*Pi*a), n = 1..infinity) HurwitzZeta[1 - s, a]=Divide[2*Gamma[s],(2*Pi)^(s)]* Sum[Divide[1,(n)^(s)]*Cos[Divide[1,2]*Pi*s - 2*n*Pi*a], {n, 1, Infinity}] Failure Failure Skip Error
25.11.E10 ${\displaystyle{\displaystyle\zeta\left(s,a\right)=\sum_{n=0}^{\infty}\frac{{% \left(s\right)_{n}}}{n!}\zeta\left(n+s\right)(1-a)^{n}}}$ Zeta(0, s, a)= sum((pochhammer(s, n))/(factorial(n))*Zeta(n + s)*(1 - a)^(n), n = 0..infinity) HurwitzZeta[s, a]= Sum[Divide[Pochhammer[s, n],(n)!]*Zeta[n + s]*(1 - a)^(n), {n, 0, Infinity}] Failure Failure Skip Error
25.11.E11 ${\displaystyle{\displaystyle\zeta\left(s,\tfrac{1}{2}\right)=(2^{s}-1)\zeta% \left(s\right)}}$ Zeta(0, s, (1)/(2))=((2)^(s)- 1)* Zeta(s) HurwitzZeta[s, Divide[1,2]]=((2)^(s)- 1)* Zeta[s] Successful Failure - Successful
25.11.E12 ${\displaystyle{\displaystyle\zeta\left(n+1,a\right)=\frac{(-1)^{n+1}{\psi^{(n)% }}\left(a\right)}{n!}}}$ Zeta(0, n + 1, a)=((- 1)^(n + 1)* subs( temp=a, diff( Psi(temp), temp$(n) ) ))/(factorial(n)) HurwitzZeta[n + 1, a]=Divide[(- 1)^(n + 1)* (D[PolyGamma[temp], {temp, n}]/.temp-> a),(n)!] Failure Failure Successful Successful 25.11.E13 ${\displaystyle{\displaystyle\zeta\left(0,a\right)=\tfrac{1}{2}-a}}$ Zeta(0, 0, a)=(1)/(2)- a HurwitzZeta[0, a]=Divide[1,2]- a Successful Successful - - 25.11.E14 ${\displaystyle{\displaystyle\zeta\left(-n,a\right)=-\frac{B_{n+1}\left(a\right% )}{n+1}}}$ Zeta(0, - n, a)= -(bernoulli(n + 1, a))/(n + 1) HurwitzZeta[- n, a]= -Divide[BernoulliB[n + 1, a],n + 1] Failure Failure Successful Successful 25.11.E15 ${\displaystyle{\displaystyle\zeta\left(s,ka\right)=k^{-s}\*\sum_{n=0}^{k-1}% \zeta\left(s,a+\frac{n}{k}\right)}}$ Zeta(0, s, k*a)= (k)^(- s)* sum(Zeta(0, s, a +(n)/(k)), n = 0..k - 1) HurwitzZeta[s, k*a]= (k)^(- s)* Sum[HurwitzZeta[s, a +Divide[n,k]], {n, 0, k - 1}] Failure Failure Skip Error 25.11.E16 ${\displaystyle{\displaystyle\zeta\left(1-s,\frac{h}{k}\right)=\frac{2\Gamma% \left(s\right)}{(2\pi k)^{s}}\*\sum_{r=1}^{k}\cos\left(\frac{\pi s}{2}-\frac{2% \pi rh}{k}\right)\zeta\left(s,\frac{r}{k}\right)}}$ Zeta(0, 1 - s, (h)/(k))=(2*GAMMA(s))/((2*Pi*k)^(s))* sum(cos((Pi*s)/(2)-(2*Pi*r*h)/(k))*Zeta(0, s, (r)/(k)), r = 1..k) HurwitzZeta[1 - s, Divide[h,k]]=Divide[2*Gamma[s],(2*Pi*k)^(s)]* Sum[Cos[Divide[Pi*s,2]-Divide[2*Pi*r*h,k]]*HurwitzZeta[s, Divide[r,k]], {r, 1, k}] Failure Failure Skip Error 25.11.E17 ${\displaystyle{\displaystyle\frac{\partial}{\partial a}\zeta\left(s,a\right)=-% s\zeta\left(s+1,a\right)}}$ diff(Zeta(0, s, a), a)= - s*Zeta(0, s + 1, a) D[HurwitzZeta[s, a], a]= - s*HurwitzZeta[s + 1, a] Successful Successful - - 25.11.E18 ${\displaystyle{\displaystyle\zeta'\left(0,a\right)=\ln\Gamma\left(a\right)-% \tfrac{1}{2}\ln\left(2\pi\right)}}$ subs( temp=0, diff( Zeta(0, temp, a), temp$(1) ) )= ln(GAMMA(a))-(1)/(2)*ln(2*Pi) (D[HurwitzZeta[temp, a], {temp, 1}]/.temp-> 0)= Log[Gamma[a]]-Divide[1,2]*Log[2*Pi] Failure Failure Successful Successful
25.11.E21 ${\displaystyle{\displaystyle\zeta'\left(1-2n,\frac{h}{k}\right)=\frac{(\psi% \left(2n\right)-\ln\left(2\pi k\right))B_{2n}\left(h/k\right)}{2n}-\frac{(\psi% \left(2n\right)-\ln\left(2\pi\right))B_{2n}}{2nk^{2n}}+\frac{(-1)^{n+1}\pi}{(2% \pi k)^{2n}}\sum_{r=1}^{k-1}\sin\left(\frac{2\pi rh}{k}\right){\psi^{(2n-1)}}% \left(\frac{r}{k}\right)+\frac{(-1)^{n+1}2\cdot(2n-1)!}{(2\pi k)^{2n}}\sum_{r=% 1}^{k-1}\cos\left(\frac{2\pi rh}{k}\right)\zeta'\left(2n,\frac{r}{k}\right)+% \frac{\zeta'\left(1-2n\right)}{k^{2n}}}}$ subs( temp=1 - 2*n, diff( Zeta(0, temp, (h)/(k)), temp$(1) ) )=((Psi(2*n)- ln(2*Pi*k))* bernoulli(2*n, h/ k))/(2*n)-((Psi(2*n)- ln(2*Pi))* bernoulli(2*n))/(2*n*(k)^(2*n))+((- 1)^(n + 1)* Pi)/((2*Pi*k)^(2*n))*sum(sin((2*Pi*r*h)/(k))*subs( temp=(r)/(k), diff( Psi(temp), temp$(2*n - 1) ) ), r = 1..k - 1)+((- 1)^(n + 1)* 2 *factorial(2*n - 1))/((2*Pi*k)^(2*n))*sum(cos((2*Pi*r*h)/(k))*subs( temp=2*n, diff( Zeta(0, temp, (r)/(k)), temp$(1) ) ), r = 1..k - 1)+(subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((k)^(2*n)) (D[HurwitzZeta[temp, Divide[h,k]], {temp, 1}]/.temp-> 1 - 2*n)=Divide[(PolyGamma[2*n]- Log[2*Pi*k])* BernoulliB[2*n, h/ k],2*n]-Divide[(PolyGamma[2*n]- Log[2*Pi])* BernoulliB[2*n],2*n*(k)^(2*n)]+Divide[(- 1)^(n + 1)* Pi,(2*Pi*k)^(2*n)]*Sum[Sin[Divide[2*Pi*r*h,k]]*(D[PolyGamma[temp], {temp, 2*n - 1}]/.temp-> Divide[r,k]), {r, 1, k - 1}]+Divide[(- 1)^(n + 1)* 2 *(2*n - 1)!,(2*Pi*k)^(2*n)]*Sum[Cos[Divide[2*Pi*r*h,k]]*(D[HurwitzZeta[temp, Divide[r,k]], {temp, 1}]/.temp-> 2*n), {r, 1, k - 1}]+Divide[D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n,(k)^(2*n)] Failure Failure Skip Error
25.11.E22 ${\displaystyle{\displaystyle\zeta'\left(1-2n,\tfrac{1}{2}\right)=-\frac{B_{2n}% \ln 2}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\zeta'\left(1-2n\right)}{2^{2n-1}}}}$ subs( temp=1 - 2*n, diff( Zeta(0, temp, (1)/(2)), temp$(1) ) )= -(bernoulli(2*n)*ln(2))/(n * (4)^(n))-(((2)^(2*n - 1)- 1)* subs( temp=1 - 2*n, diff( Zeta(temp), temp$(1) ) ))/((2)^(2*n - 1)) (D[HurwitzZeta[temp, Divide[1,2]], {temp, 1}]/.temp-> 1 - 2*n)= -Divide[BernoulliB[2*n]*Log[2],n * (4)^(n)]-Divide[((2)^(2*n - 1)- 1)* (D[Zeta[temp], {temp, 1}]/.temp-> 1 - 2*n),(2)^(2*n - 1)] Failure Failure Successful Successful
25.11.E23