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A175571 Decimal expansion of the Dirichlet beta function of 5. 8
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 (list; constant; graph; refs; listen; history; text; internal format)
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

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]

Also equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - Jean-François Alcover, Mar 29 2013

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

Cf. A003881 (at 1), A006752 (at 2), A153071 (at 3), A175572 (at 4), A175570 (at 6).

Sequence in context: A136130 A144668 A021505 * A019894 A262823 A157293

Adjacent sequences:  A175568 A175569 A175570 * A175572 A175573 A175574

KEYWORD

cons,easy,nonn

AUTHOR

R. J. Mathar, Jul 15 2010

STATUS

approved

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Last modified August 16 03:24 EDT 2018. Contains 313782 sequences. (Running on oeis4.)