A324993 Decimal expansion of zeta'(-1, 1/3).

0, 9, 3, 7, 2, 6, 2, 0, 1, 7, 6, 0, 7, 7, 9, 4, 2, 7, 4, 8, 4, 2, 0, 0, 8, 9, 9, 1, 3, 3, 1, 9, 2, 8, 6, 7, 3, 6, 8, 8, 3, 7, 2, 8, 6, 9, 3, 8, 7, 3, 8, 0, 2, 1, 5, 2, 5, 4, 4, 8, 0, 9, 2, 5, 4, 5, 4, 3, 4, 9, 9, 7, 9, 5, 0, 9, 2, 3, 3, 5, 1, 1, 7, 1, 6, 7, 2, 7, 4, 9, 4, 7, 5, 5, 4, 0, 7, 6, 0, 4, 0, 2, 9, 8, 5, 1

Offset: 0

Author

Vaclav Kotesovec, Mar 23 2019

Keywords

• nonn
• cons

Examples

			0.093726201760779427484200899133192867368837286938738021525448092545434...
		

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 23

Crossrefs

Cf. A084448, A240966, A306651, A324994.

Programs

Maple

evalf(Zeta(1,-1,1/3), 120);
evalf(-Pi/(18*sqrt(3)) - log(3)/72 + Psi(1, 1/3) / (12*sqrt(3)*Pi) - Zeta(1, -1)/3, 120);

Mathematica

RealDigits[Derivative[1, 0][Zeta][-1, 1/3], 10, 120][[1]]
N[With[{k=1}, -Sqrt[3] * (9^k - 1) * BernoulliB[2*k] * Pi / (9^k * 8*k) - 3*BernoulliB[2*k] * Log[3] / 9^k / 4 / k - (-1)^k * PolyGamma[2*k-1, 1/3] / 2 / Sqrt[3] / (6*Pi)^(2*k-1) - (9^k-3)*Zeta'[-2*k+1]/2/9^k], 120]

PARI

zetahurwitz'(-1, 1/3) \\ Michel Marcus, Mar 24 2019

Formula

Equals -Pi/(18*sqrt(3)) - log(3)/72 + PolyGamma(1, 1/3) / (12*sqrt(3)*Pi) - Zeta'(-1)/3.

A324993 + A324994 = -log(3)/36 - 2*Zeta'(-1)/3.