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A175570 Decimal expansion of the Dirichlet beta function of 6. 9
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 (list; constant; graph; refs; listen; history; text; internal format)
OFFSET

0,1

REFERENCES

L. B. W. Jolley, Summation of Series, Dover (1971) 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.

Wikipedia, Dirichlet beta function.

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]

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

EXAMPLE

0.998685222218438135441600...

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 *)

PROG

(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

CROSSREFS

Cf. A003881 (at 1), A006752 (at 2), A153071 (at 3), A175572 (at 4), A175571 (at 5).

Sequence in context: A346438 A304029 A019896 * A050812 A139345 A231470

Adjacent sequences:  A175567 A175568 A175569 * A175571 A175572 A175573

KEYWORD

cons,easy,nonn

AUTHOR

R. J. Mathar, Jul 15 2010

STATUS

approved

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Last modified August 5 14:42 EDT 2021. Contains 346469 sequences. (Running on oeis4.)