A324994 Decimal expansion of zeta'(-1, 2/3) (negated).

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

Offset

0,3

Links

Table of n, a(n) for n=0..105.

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

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.

Example

-0.01396244731237074388034460444140926398857665998812431718484139757490...

Maple

evalf(Zeta(1, -1, 2/3), 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, 2/3], 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]

Prog

(PARI) zetahurwitz'(-1, 2/3) \\ Michel Marcus, Mar 24 2019

Crossrefs

Cf. A084448, A240966, A324993.

Sequence in context: A020834 A198731 A153019 * A201614 A021720 A371752

Adjacent sequences: A324991 A324992 A324993 * A324995 A324996 A324997

Keyword

nonn,cons

Author

Vaclav Kotesovec, Mar 23 2019

Status

approved