Formula:DLMF:25.11:E23

<< Formula:DLMF:25.11:E22

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E24 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\tfrac{1}{3}}=-\frac{\pi(9^{n}-1)\BernoulliB{2n}}{8n\sqrt{3}(3^{2n-1}-1)}-\frac{\BernoulliB{2n}\ln 3}{4n\cdot 3^{2n-1}}-\frac{(-1)^{n}{\psi^{(2n-1)}}\left(\frac{1}{3}\right)}{2\sqrt{3}(6\pi)^{2n-1}}-\frac{\left(3^{2n-1}-1\right)\RiemannZeta^{\prime}@{1-2n}}{2\cdot 3^{2n-1}}}}}$

Constraint(s)

${\displaystyle{\displaystyle{\displaystyle n=1,2,3,\dots}}}$

Note(s)

primes on ${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta}}}$ denote derivatives with respect to ${\displaystyle{\displaystyle{\displaystyle s}}}$

Proof

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Symbols List

${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Hurwitz zeta function : http://dlmf.nist.gov/25.11#E1
${\displaystyle{\displaystyle{\displaystyle\pi}}}$ : ratio of a circle's circumference to its diameter : http://dlmf.nist.gov/5.19.E4
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : Bernoulli polynomial : http://dlmf.nist.gov/24.2#i
${\displaystyle{\displaystyle{\displaystyle\mathrm{ln}}}}$ : principal branch of logarithm function : http://dlmf.nist.gov/4.2#E2
${\displaystyle{\displaystyle{\displaystyle(-1)}}}$ : negative unity to an integer power : http://dlmf.nist.gov/5.7.E7
${\displaystyle{\displaystyle{\displaystyle\psi}}}$ : psi (or digamma) function : http://dlmf.nist.gov/5.2#E2
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

Bibliography

Equation (23), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E22

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E24 >>