A175570 Decimal expansion of the Dirichlet beta function of 6.

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

Offset: 0

Author

R. J. Mathar, Jul 15 2010

Keywords

• cons
• easy
• nonn

Examples

			0.998685222218438135441600...
		

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)), A175571 (beta(5)), 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(6) ; x := evalf(x) ;

Mathematica

RealDigits[ DirichletBeta[6], 10, 105] // First (* Jean-François Alcover, Feb 11 2013, updated Mar 14 2018 *)

PARI

beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
beta(6) \\ Charles R Greathouse IV, Jan 31 2018

PARI

sumpos(n=1,(12288*n^5 - 30720*n^4 + 33280*n^3 - 19200*n^2 + 5808*n - 728)/(16777216*n^12 - 100663296*n^11 + 270532608*n^10 - 429916160*n^9 + 449249280*n^8 - 324796416*n^7 + 166445056*n^6 - 60899328*n^5 + 15793920*n^4 - 2833920*n^3 + 334368*n^2 - 23328*n + 729),1) \\ Charles R Greathouse IV, Feb 01 2018

Formula

Equals Sum_{n>=1} A101455(n)/n^6. [see arxiv:1008.2547, L(m=4,r=2,s=6)] [corrected by R. J. Mathar, Feb 01 2018]

Equals (PolyGamma(5, 1/4) - PolyGamma(5, 3/4))/491520. - Jean-François Alcover, Jun 11 2015

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^6)^(-1). - Amiram Eldar, Nov 06 2023