A259070 Decimal expansion of zeta'(-5) (the derivative of Riemann's zeta function at -5) (negated).

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

Offset: 0

Author

Jean-François Alcover, Jun 18 2015

Keywords

• nonn
• cons

Examples

			-0.000572985980198635204990994148833874513253987291199521217820791880997735...
		

References

• Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.15.1 Generalized Glaisher constants, p. 136-137.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..10000
• J. B. Rosser, L. Schoenfeld, Approximate formulas for some functions of prime numbers, Ill. J. Math. 6 (1) (1962) 64-94, Table IV
• Eric Weisstein's MathWorld, Riemann Zeta Function.
• Wikipedia, Riemann Zeta Function
• Index entries for constants related to zeta

Crossrefs

Cf. A000417, A000428, A023874, A260404.

Programs

Mathematica

Join[{0, 0, 0}, RealDigits[Zeta'[-5], 10, 103] // First]

Formula

zeta'(-n) = (BernoulliB(n+1)*HarmonicNumber(n))/(n+1) - log(A(n)), where A(n) is the n-th Bendersky constant, that is the n-th generalized Glaisher constant.

zeta'(-5) = 137/15120 - log(A(5)), where A(5) is A243265.

Equals 137/15120 - (gamma + log(2*Pi))/252 + 15*Zeta'(6) / (4*Pi^6), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jul 25 2015