OFFSET
0,1
COMMENTS
The value of the Dirichlet L-series L(m=4,r=2,s=4), see arXiv:1008.2547.
REFERENCES
L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308.
LINKS
G. C. Greubel, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.
FORMULA
Equals 5*Pi^5/1536 = Sum_{n>=1} A101455(n)/n^5, where Pi^5 = A092731. [corrected by R. J. Mathar, Feb 01 2018]
Equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - Jean-François Alcover, Mar 29 2013
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^5)^(-1). - Amiram Eldar, Nov 06 2023
EXAMPLE
0.99615782807708806400631936...
MAPLE
DirichletBeta := proc(s) 4^(-s)*(Zeta(0, s, 1/4)-Zeta(0, s, 3/4)) ; end proc: x := DirichletBeta(5) ; x := evalf(x) ;
MATHEMATICA
RealDigits[ DirichletBeta[5], 10, 105] // First (* Jean-François Alcover, Feb 20 2013, updated Mar 14 2018 *)
PROG
(PARI) 5*Pi^5/1536 \\ Charles R Greathouse IV, Jan 31 2018
(PARI) beta(x)=(zetahurwitz(x, 1/4)-zetahurwitz(x, 3/4))/4^x
beta(5) \\ Charles R Greathouse IV, Jan 31 2018
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Jul 15 2010
STATUS
approved