A324995 Decimal expansion of zeta'(-1, 1/4).
0, 9, 3, 5, 6, 7, 8, 6, 8, 9, 7, 0, 2, 6, 1, 0, 6, 1, 1, 8, 6, 3, 3, 6, 0, 7, 1, 6, 4, 7, 4, 4, 6, 3, 1, 0, 0, 6, 1, 5, 2, 1, 0, 8, 6, 0, 3, 8, 3, 5, 9, 5, 4, 0, 5, 2, 3, 5, 6, 5, 6, 8, 0, 5, 7, 2, 6, 0, 6, 8, 7, 1, 6, 7, 8, 4, 3, 1, 8, 6, 2, 0, 2, 6, 5, 9, 7, 3, 4, 3, 6, 1, 7, 3, 4, 7, 1, 0, 9, 1, 6, 9, 5, 4, 0, 3
Offset: 0
Examples
0.093567868970261061186336071647446310061521086038359540523565680572606...
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.
Programs
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Maple
evalf(Zeta(1,-1,1/4), 120); evalf(-Pi/32 + Psi(1, 1/4)/(32*Pi) - Zeta(1,-1)/8, 120);
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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] -
PARI
zetahurwitz'(-1, 1/4) \\ Michel Marcus, Mar 24 2019
Formula
Equals -Pi/32 + PolyGamma(1, 1/4)/(32*Pi) - Zeta'(-1)/8.