Formula:DLMF:25.11:E22

<< Formula:DLMF:25.11:E21

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E23 >>

${\displaystyle{\displaystyle{\displaystyle\HurwitzZeta^{\prime}@{1-2n}{\tfrac{1}{2}}=-\frac{\BernoulliB{2n}\ln 2}{n\cdot 4^{n}}-\frac{(2^{2n-1}-1)\RiemannZeta^{\prime}@{1-2n}}{2^{2n-1}}}}}$

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
${\displaystyle{\displaystyle{\displaystyle B_{n}}}}$ : 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
${\displaystyle{\displaystyle{\displaystyle\zeta}}}$ : Riemann zeta function : http://dlmf.nist.gov/25.2#E1

Bibliography

Equation (22), Section 25.11 of DLMF.

URL links

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<< Formula:DLMF:25.11:E21

formula in Hurwitz Zeta Function

Formula:DLMF:25.11:E23 >>