A078127 Decimal expansion of DirichletBeta'(1).
1, 9, 2, 9, 0, 1, 3, 1, 6, 7, 9, 6, 9, 1, 2, 4, 2, 9, 3, 6, 3, 1, 8, 9, 7, 6, 4, 0, 2, 8, 0, 3, 2, 7, 8, 5, 2, 4, 5, 0, 9, 6, 8, 6, 7, 6, 2, 0, 0, 0, 7, 5, 2, 7, 1, 7, 1, 3, 4, 9, 2, 2, 7, 4, 4, 3, 6, 0, 5, 7, 0, 3, 5, 9, 2, 7, 7, 8, 7, 7, 0, 3, 9, 1, 4, 4, 3, 0, 5, 5, 1, 6, 3, 8, 7, 8, 4, 6, 0, 4, 7
Offset: 0
Examples
0.1929013167969124293631897640...
Links
- Steven R. Finch, Errata and Addenda to Mathematical Constants. p. 8.
- Eric Weisstein's World of Mathematics, Dirichlet Beta Function
Programs
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Maple
Pi/4*(gamma+log(2*Pi)-2*log(GAMMA(1/4)/GAMMA(3/4))); evalf(%) ; # R. J. Mathar, Jun 10 2024
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Mathematica
Prepend@@RealDigits[(Pi*(EulerGamma + 2*Log[2] + 3*Log[Pi] - 4*Log[Gamma[1/4]]))/4, 10, 101]
Formula
Equals (Pi/4)*(gamma + log(2*Pi) - 2*log(Gamma(1/4)/Gamma(3/4))), where gamma is Euler's constant and Gamma(x) is the Euler Gamma function.
Equals Sum_{k>=1} (-1)^(k+1)*log(2*k+1)/(2*k+1). - Jean-François Alcover, Aug 11 2014