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Decimal expansion of beta'(0) : the derivative of the Dirichlet beta function evaluated at zero.
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%I #20 Oct 11 2024 16:11:30

%S 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,

%T 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,

%U 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

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

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/DirichletBetaFunction.html">Dirichlet Beta Function</a>

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

%F 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

%F Equals Sum_{n>=1} (-1)^(n+1) * arccoth(2*n). - _Antonio Graciá Llorente_, Oct 11 2024

%F Equals log(Gamma(1/4)^2 / (2^(3/2)*Pi)). - _Vaclav Kotesovec_, Oct 11 2024

%e 0.39159439270683677647194534689911102809021011577...

%t RealDigits[ Log[ (2*Gamma[5/4])/Gamma[3/4]], 10, 104] // First (* _Jean-François Alcover_, Feb 20 2013 *)

%K cons,nonn

%O 0,1

%A _Benoit Cloitre_, Jan 24 2006