A175571 Decimal expansion of the Dirichlet beta function of 5.

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

Offset: 0

Author

R. J. Mathar, Jul 15 2010

Keywords

• cons
• easy
• nonn

Comments

The value of the Dirichlet L-series L(m=4,r=2,s=4), see arXiv:1008.2547.

Examples

			0.99615782807708806400631936...
		

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.

Crossrefs

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

Programs

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 *)

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

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