# Formula:DLMF:25.11:E23

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

## Constraint(s)

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

## Note(s)

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

## Proof

We ask users to provide proof(s), reference(s) to proof(s), or further clarification on the proof(s) in this space.

## Symbols List

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

## URL links

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