Integral Representations

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Integral Representations

In Terms of Elementary Functions

\RiemannZeta @ s = 1 Γ ( s ) 0 x s - 1 e x - 1 d x \RiemannZeta @ 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑥 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{\Gamma% \left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{{\mathrm{e}^{x}}-1}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x-1} \diff{x} }

Constraint(s): s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}


\RiemannZeta @ s = 1 Γ ( s + 1 ) 0 e x x s ( e x - 1 ) 2 d x \RiemannZeta @ 𝑠 1 Euler-Gamma 𝑠 1 superscript subscript 0 𝑥 superscript 𝑥 𝑠 superscript 𝑥 1 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{\Gamma% \left(s+1\right)}\int_{0}^{\infty}\frac{{\mathrm{e}^{x}}x^{s}}{({\mathrm{e}^{x% }}-1)^{2}}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{\EulerGamma@{s+1}} \int_0^\infty \frac{\expe^x x^s}{(\expe^x-1)^2} \diff{x} }

Constraint(s): s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}


\RiemannZeta @ s = 1 ( 1 - 2 1 - s ) Γ ( s ) 0 x s - 1 e x + 1 d x \RiemannZeta @ 𝑠 1 1 superscript 2 1 𝑠 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑥 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{(1-2^{1-s}% )\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{{\mathrm{e}^{x}}+1}% \mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{(1 - 2^{1-s}) \EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\expe^x+1} \diff{x} }

Constraint(s): s > 0 𝑠 0 {\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}} &
s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ s = 1 ( 1 - 2 1 - s ) Γ ( s + 1 ) 0 e x x s ( e x + 1 ) 2 d x \RiemannZeta @ 𝑠 1 1 superscript 2 1 𝑠 Euler-Gamma 𝑠 1 superscript subscript 0 𝑥 superscript 𝑥 𝑠 superscript 𝑥 1 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{(1-2^{1-s}% )\Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{{\mathrm{e}^{x}}x^{s}}{({% \mathrm{e}^{x}}+1)^{2}}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{(1 - 2^{1-s}) \EulerGamma@{s+1}} \int_0^\infty \frac{\expe^x x^s}{(\expe^x+1)^2} \diff{x} }

Constraint(s): s > 0 𝑠 0 {\displaystyle{\displaystyle{\displaystyle\Re{s}>0}}} &
s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ s = - s 0 x - x - 1 2 x s + 1 d x \RiemannZeta @ 𝑠 𝑠 superscript subscript 0 𝑥 𝑥 1 2 superscript 𝑥 𝑠 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=-s\int_{0}^{\infty}% \frac{x-\left\lfloor x\right\rfloor-\frac{1}{2}}{x^{s+1}}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = -s \int_0^\infty \frac{x-\floor{x}-\frac{1}{2}}{x^{s+1}} \diff{x} }

Constraint(s): - 1 < s < 0 1 𝑠 0 {\displaystyle{\displaystyle{\displaystyle-1<\Re{s}<0}}}


\RiemannZeta @ s = 1 2 + 1 s - 1 + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 ) x s - 1 e x d x \RiemannZeta @ 𝑠 1 2 1 𝑠 1 1 Euler-Gamma 𝑠 superscript subscript 0 1 𝑥 1 1 𝑥 1 2 superscript 𝑥 𝑠 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{2}+\frac{1% }{s-1}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}\left(\frac{1}{{\mathrm{% e}^{x}}-1}-\frac{1}{x}+\frac{1}{2}\right)\frac{x^{s-1}}{{\mathrm{e}^{x}}}% \mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} \right) \frac{x^{s-1}}{\expe^x} \diff{x} }

Constraint(s): s > - 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-1}}} &
s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ s = 1 2 + 1 s - 1 + m = 1 n \BernoulliB 2 m ( 2 m ) ! Γ ( s + 2 m - 1 ) Γ ( s ) + 1 Γ ( s ) 0 ( 1 e x - 1 - 1 x + 1 2 - m = 1 n \BernoulliB 2 m ( 2 m ) ! x 2 m - 1 ) x s - 1 e x d x \RiemannZeta @ 𝑠 1 2 1 𝑠 1 superscript subscript 𝑚 1 𝑛 \BernoulliB 2 𝑚 2 𝑚 Euler-Gamma 𝑠 2 𝑚 1 Euler-Gamma 𝑠 1 Euler-Gamma 𝑠 superscript subscript 0 1 𝑥 1 1 𝑥 1 2 superscript subscript 𝑚 1 𝑛 \BernoulliB 2 𝑚 2 𝑚 superscript 𝑥 2 𝑚 1 superscript 𝑥 𝑠 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{2}+\frac{1% }{s-1}+\sum_{m=1}^{n}\frac{\BernoulliB{2m}}{(2m)!}\frac{\Gamma\left(s+2m-1% \right)}{\Gamma\left(s\right)}+\frac{1}{\Gamma\left(s\right)}\int_{0}^{\infty}% \left(\frac{1}{{\mathrm{e}^{x}}-1}-\frac{1}{x}+\frac{1}{2}-\sum_{m=1}^{n}\frac% {\BernoulliB{2m}}{(2m)!}x^{2m-1}\right)\frac{x^{s-1}}{{\mathrm{e}^{x}}}\mathrm% {d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + \sum_{m=1}^n \frac{\BernoulliB{2m}}{(2m)!} \frac{\EulerGamma@{s+2m-1}}{\EulerGamma@{s}} + \frac{1}{\EulerGamma@{s}} \int_0^\infty \left( \frac{1}{\expe^x-1} - \frac{1}{x} + \frac{1}{2} - \sum_{m=1}^n \frac{\BernoulliB{2m}}{(2m)!} x^{2m-1} \right) \frac{x^{s-1}}{\expe^x} \diff{x} }

Constraint(s): s > - ( 2 n + 1 ) 𝑠 2 𝑛 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>-(2n+1)}}} &
n = 1 , 2 , 3 , 𝑛 1 2 3 {\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}} &
s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ s = 1 2 ( 1 - 2 - s ) Γ ( s ) 0 x s - 1 sinh x d x \RiemannZeta @ 𝑠 1 2 1 superscript 2 𝑠 Euler-Gamma 𝑠 superscript subscript 0 superscript 𝑥 𝑠 1 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{2(1-2^{-s}% )\Gamma\left(s\right)}\int_{0}^{\infty}\frac{x^{s-1}}{\sinh x}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{2 (1 - 2^{-s}) \EulerGamma@{s}} \int_0^\infty \frac{x^{s-1}}{\sinh@@{x}} \diff{x} }

Constraint(s): s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}


\RiemannZeta @ s = 2 s - 1 Γ ( s + 1 ) 0 x s ( sinh x ) 2 d x \RiemannZeta @ 𝑠 superscript 2 𝑠 1 Euler-Gamma 𝑠 1 superscript subscript 0 superscript 𝑥 𝑠 superscript 𝑥 2 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{2^{s-1}}{% \Gamma\left(s+1\right)}\int_{0}^{\infty}\frac{x^{s}}{(\sinh x)^{2}}\mathrm{d}x% }}} {\displaystyle \RiemannZeta@{s} = \frac{2^{s-1}}{\EulerGamma@{s+1}} \int_0^\infty \frac{x^s}{(\sinh@@{x})^2} \diff{x} }

Constraint(s): s > 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle\Re{s}>1}}}


\RiemannZeta @ s = 2 s - 1 1 - 2 1 - s 0 cos ( s arctan x ) ( 1 + x 2 ) s / 2 cosh ( 1 2 π x ) d x \RiemannZeta @ 𝑠 superscript 2 𝑠 1 1 superscript 2 1 𝑠 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 1 2 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\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}}} {\displaystyle \RiemannZeta@{s} = \frac{2^{s-1}}{1 - 2^{1-s}} \int_0^\infty \frac{\cos@{s \atan@@{x}}}{(1 + x^2)^{s/2} \cosh@{\frac{1}{2} \cpi x}} \diff{x} }

Constraint(s): s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ s = 1 2 + 1 s - 1 + 2 0 sin ( s arctan x ) ( 1 + x 2 ) s / 2 ( e 2 π x - 1 ) d x \RiemannZeta @ 𝑠 1 2 1 𝑠 1 2 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 2 𝑥 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{1}{2}+\frac{1% }{s-1}+2\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^% {2})^{s/2}({\mathrm{e}^{2\pi x}}-1)}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{2} + \frac{1}{s-1} + 2 \int_0^\infty \frac{\sin@{s \atan@@{x}}}{(1 + x^2)^{s/2} (\expe^{2\cpi x} - 1)} \diff{x} }

Constraint(s): s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ s = 2 s - 1 s - 1 - 2 s 0 sin ( s arctan x ) ( 1 + x 2 ) s / 2 ( e π x + 1 ) d x \RiemannZeta @ 𝑠 superscript 2 𝑠 1 𝑠 1 superscript 2 𝑠 superscript subscript 0 𝑠 𝑥 superscript 1 superscript 𝑥 2 𝑠 2 𝑥 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{2^{s-1}}{s-1}% -2^{s}\int_{0}^{\infty}\frac{\sin\left(s\operatorname{arctan}x\right)}{(1+x^{2% })^{s/2}({\mathrm{e}^{\pi x}}+1)}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{s} = \frac{2^{s-1}}{s-1} - 2^s \int_0^\infty \frac{\sin@{s \atan@@{x}}}{(1 + x^2)^{s/2} (\expe^{\cpi x} + 1)} \diff{x} }

Constraint(s): s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


In Terms of Other Functions

\RiemannZeta @ s = π s / 2 s ( s - 1 ) Γ ( 1 2 s ) + π s / 2 Γ ( 1 2 s ) 1 ( x s / 2 + x ( 1 - s ) / 2 ) ω ( x ) x d x \RiemannZeta @ 𝑠 𝑠 2 𝑠 𝑠 1 Euler-Gamma 1 2 𝑠 𝑠 2 Euler-Gamma 1 2 𝑠 superscript subscript 1 superscript 𝑥 𝑠 2 superscript 𝑥 1 𝑠 2 𝜔 𝑥 𝑥 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\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}}} {\displaystyle \RiemannZeta@{s} = \frac{\cpi^{s/2}}{s(s-1)\EulerGamma@{\frac{1}{2}s}} + \frac{\cpi^{s/2}}{\EulerGamma@{\frac{1}{2}s}} \* \int_1^\infty \left (x^{s/2} + x^{(1-s)/2} \right) \frac{\omega(x)}{x} \diff{x} }

Substitution(s): ω ( x ) = n = 1 e - n 2 π x = 1 2 ( \JacobiThetaTau 3 @ 0 i x - 1 ) 𝜔 𝑥 superscript subscript 𝑛 1 superscript 𝑛 2 𝑥 1 2 \JacobiThetaTau 3 @ 0 imaginary-unit 𝑥 1 {\displaystyle{\displaystyle{\displaystyle{\displaystyle\omega(x)=\sum_{n=1}^{% \infty}{\mathrm{e}^{-n^{2}\pi x}}=\frac{1}{2}\left(\JacobiThetaTau{3}@{0}{% \mathrm{i}x}-1\right)}}}}


Constraint(s): s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ s = 1 s - 1 + sin ( π s ) π 0 ( ln ( 1 + x ) - ψ ( 1 + x ) ) x - s d x \RiemannZeta @ 𝑠 1 𝑠 1 𝑠 superscript subscript 0 1 𝑥 digamma 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\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}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \frac{\sin@{\cpi s}}{\cpi} \* \int_0^\infty (\ln@{1+x} - \digamma@{1+x}) x^{-s} \diff{x} }

Constraint(s): 0 < s < 1 0 𝑠 1 {\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}


\RiemannZeta @ s = 1 s - 1 + sin ( π s ) π ( s - 1 ) 0 ( 1 1 + x - ψ ( 1 + x ) ) x 1 - s d x \RiemannZeta @ 𝑠 1 𝑠 1 𝑠 𝑠 1 superscript subscript 0 1 1 𝑥 diffop digamma 1 1 𝑥 superscript 𝑥 1 𝑠 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\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}}} {\displaystyle \RiemannZeta@{s} = \frac{1}{s-1} + \frac{\sin@{\cpi s}}{\cpi (s-1)} \* \int_0^\infty \left( \frac{1}{1+x} - \digamma'@{1+x} \right) x^{1-s} \diff{x} }

Constraint(s): 0 < s < 2 0 𝑠 2 {\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<2}}} , s 1 𝑠 1 {\displaystyle{\displaystyle{\displaystyle s\neq 1}}}


\RiemannZeta @ 1 + s = sin ( π s ) π 0 ( γ + ψ ( 1 + x ) ) x - s - 1 d x \RiemannZeta @ 1 𝑠 𝑠 superscript subscript 0 digamma 1 𝑥 superscript 𝑥 𝑠 1 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{1+s}=\frac{\sin\left(% \pi s\right)}{\pi}\int_{0}^{\infty}\left(\gamma+\psi\left(1+x\right)\right)x^{% -s-1}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{1+s} = \frac{\sin@{\cpi s}}{\cpi} \int_0^\infty \left( \EulerConstant + \digamma@{1+x} \right) x^{-s-1} \diff{x} }

Constraint(s): 0 < s < 1 0 𝑠 1 {\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}


\RiemannZeta @ 1 + s = sin ( π s ) π s 0 ψ ( 1 + x ) x - s d x \RiemannZeta @ 1 𝑠 𝑠 𝑠 superscript subscript 0 diffop digamma 1 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{1+s}=\frac{\sin\left(% \pi s\right)}{\pi s}\int_{0}^{\infty}\psi'\left(1+x\right)x^{-s}\mathrm{d}x}}} {\displaystyle \RiemannZeta@{1+s} = \frac{\sin@{\cpi s}}{\cpi s} \int_0^\infty \digamma'@{1+x} x^{-s} \diff{x} }

Constraint(s): 0 < s < 1 0 𝑠 1 {\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}


\RiemannZeta @ m + s = ( - 1 ) m - 1 Γ ( s ) sin ( π s ) π Γ ( m + s ) 0 ψ ( m ) ( 1 + x ) x - s d x \RiemannZeta @ 𝑚 𝑠 superscript 1 𝑚 1 Euler-Gamma 𝑠 𝑠 Euler-Gamma 𝑚 𝑠 superscript subscript 0 digamma 𝑚 1 𝑥 superscript 𝑥 𝑠 𝑥 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{m+s}=(-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}}} {\displaystyle \RiemannZeta@{m+s} = \opminus^{m-1} \frac{\EulerGamma@{s} \sin@{\cpi s}}{\cpi \EulerGamma@{m+s}} \* \int_0^\infty \digamma^{(m)}@{1+x} x^{-s} \diff{x} }

Constraint(s): m = 1 , 2 , 3 , 𝑚 1 2 3 {\displaystyle{\displaystyle{\displaystyle m=1,2,3,\dots}}} &
0 < s < 1 0 𝑠 1 {\displaystyle{\displaystyle{\displaystyle 0<\Re{s}<1}}}


Contour Integrals

\RiemannZeta @ s = Γ ( 1 - s ) 2 π i - ( 0 + ) z s - 1 e - z - 1 d z \RiemannZeta @ 𝑠 Euler-Gamma 1 𝑠 2 imaginary-unit superscript subscript limit-from 0 superscript 𝑧 𝑠 1 𝑧 1 𝑧 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{\Gamma\left(1% -s\right)}{2\pi\mathrm{i}}\int_{-\infty}^{(0+)}\frac{z^{s-1}}{{\mathrm{e}^{-z}% }-1}\mathrm{d}z}}} {\displaystyle \RiemannZeta@{s} = \frac{\EulerGamma@{1-s}}{2 \cpi \iunit} \int_{-\infty}^{(0+)} \frac{z^{s-1}}{\expe^{-z}-1} \diff{z} }

Constraint(s): s 1 , 2 , 𝑠 1 2 {\displaystyle{\displaystyle{\displaystyle s\neq 1,2,\dots}}} &

The integration contour is a loop around the negative real axis; it starts at - {\displaystyle{\displaystyle{\displaystyle-\infty}}} , encircles the origin once in the positive direction without enclosing any of the points

z = ± 2 π i 𝑧 plus-or-minus 2 imaginary-unit {\displaystyle{\displaystyle{\displaystyle z=\pm 2\pi\mathrm{i}}}} , ± 4 π i , , plus-or-minus 4 imaginary-unit {\displaystyle{\displaystyle{\displaystyle\pm 4\pi\mathrm{i},\ldots,}}} and returns to - {\displaystyle{\displaystyle{\displaystyle-\infty}}}


\RiemannZeta @ s = Γ ( 1 - s ) 2 π i ( 1 - 2 1 - s ) - ( 0 + ) z s - 1 e - z + 1 d z \RiemannZeta @ 𝑠 Euler-Gamma 1 𝑠 2 imaginary-unit 1 superscript 2 1 𝑠 superscript subscript limit-from 0 superscript 𝑧 𝑠 1 𝑧 1 𝑧 {\displaystyle{\displaystyle{\displaystyle\RiemannZeta@{s}=\frac{\Gamma\left(1% -s\right)}{2\pi\mathrm{i}(1-2^{1-s})}\*\int_{-\infty}^{(0+)}\frac{z^{s-1}}{{% \mathrm{e}^{-z}}+1}\mathrm{d}z}}} {\displaystyle \RiemannZeta@{s} = \frac{\EulerGamma@{1-s}}{2 \cpi \iunit (1 - 2^{1-s})} \* \int_{-\infty}^{(0+)} \frac{z^{s-1}}{\expe^{-z}+1} \diff{z} }

Constraint(s): s 1 , 2 , 𝑠 1 2 {\displaystyle{\displaystyle{\displaystyle s\neq 1,2,\dots}}} &

The contour here is any loop that encircles the origin in the positive direction

not enclosing any of the points ± π i plus-or-minus imaginary-unit {\displaystyle{\displaystyle{\displaystyle\pm\pi\mathrm{i}}}} , ± 3 π i , plus-or-minus 3 imaginary-unit {\displaystyle{\displaystyle{\displaystyle\pm 3\pi\mathrm{i},\ldots}}}