A259069 Decimal expansion of zeta'(-4) (the derivative of Riemann's zeta function at -4).

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

Offset: 0

Author

Jean-François Alcover, Jun 18 2015

Keywords

• nonn
• cons

Examples

			0.00798381145026862428069667079878930390523769336229887641770473971402874...
		

References

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

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1500
• Eric Weisstein's MathWorld, Riemann Zeta Function.
• Wikipedia, Riemann Zeta Function
• Index entries for constants related to zeta

Programs

Mathematica

Join[{0, 0}, RealDigits[Zeta'[-4], 10, 104] // 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'(-4) = 3*zeta(5)/(4*Pi^4) = -log(A(4)), where A(4) is A243264.