OFFSET
0,2
LINKS
J. Miller and V. Adamchik, Derivatives of the Hurwitz Zeta Function for Rational Arguments, Journal of Computational and Applied Mathematics 100 (1998) 201-206. [contains a large number of typos]
Eric Weisstein's World of Mathematics, Hurwitz Zeta Function, formula 24.
FORMULA
Equals -Pi/32 + PolyGamma(1, 1/4)/(32*Pi) - Zeta'(-1)/8.
EXAMPLE
0.093567868970261061186336071647446310061521086038359540523565680572606...
MAPLE
evalf(Zeta(1, -1, 1/4), 120);
evalf(-Pi/32 + Psi(1, 1/4)/(32*Pi) - Zeta(1, -1)/8, 120);
MATHEMATICA
RealDigits[Derivative[1, 0][Zeta][-1, 1/4], 10, 120][[1]]
N[With[{k=1}, -(4^k-1) * BernoulliB[2*k] * Pi / 4^(k+1)/k + (4^(k-1)-1)*BernoulliB[2*k] * Log[2]/k/2^(4*k-1) - (-1)^k*PolyGamma[2*k-1, 1/4] / 4 / (8*Pi)^(2*k-1) - (4^k - 2)*Zeta'[1-2*k]/2^(4*k)], 120]
PROG
(PARI) zetahurwitz'(-1, 1/4) \\ Michel Marcus, Mar 24 2019
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Vaclav Kotesovec, Mar 23 2019
STATUS
approved