A113847 Decimal expansion of beta'(0) : the derivative of the Dirichlet beta function evaluated at zero.

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

Offset

0,1

Links

Table of n, a(n) for n=0..103.

Eric Weisstein's World of Mathematics, Dirichlet Beta Function

Formula

beta'(0)=log(Gamma(1/4)/2/Gamma(3/4))=0.3915943927068367764719453...

Equals Sum_{n>=1} (1/2)*log((16*(2*n + 1)*(2*n - 1)^3)/((4*n - 3)*(4*n + 1))^2). - Antonio Graciá Llorente, Sep 05 2024

Equals Sum_{n>=1} (-1)^(n+1) * arccoth(2*n). - Antonio Graciá Llorente, Oct 11 2024

Equals log(Gamma(1/4)^2 / (2^(3/2)*Pi)). - Vaclav Kotesovec, Oct 11 2024

Example

0.39159439270683677647194534689911102809021011577...

Mathematica

RealDigits[ Log[ (2*Gamma[5/4])/Gamma[3/4]], 10, 104] // First (* Jean-François Alcover, Feb 20 2013 *)

Crossrefs

Sequence in context: A019665 A119898 A371497 * A119796 A154572 A333346

Adjacent sequences: A113844 A113845 A113846 * A113848 A113849 A113850

Keyword

cons,nonn

Author

Benoit Cloitre, Jan 24 2006

Status

approved