A153071 Decimal expansion of L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.

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

Offset

0,1

References

Leonhard Euler, Introductio in Analysin Infinitorum, First Part, Articles 175, 284 and 287.

Bruce C. Berndt, Ramanujan's Notebooks, Part II, Springer-Verlag, 1989. See page 293, Entry 25 (iii).

Links

Table of n, a(n) for n=0..89.

J. T. Groenman, Problem 1511, Crux Mathematicorum, Vol. 16, No. 2 (1990), p. 43; Solution to Problem 1511, by Beatriz Margolis, ibid., Vol. 17, No. 3 (1991), pp. 92-93.

Qing-Hu Hou and Zhi-Wei Sun, A q-analogue of the identity Sum_{k>=0}(-1)^k/(2k+1)^3 = Pi^3/32, arXiv:1808.04717 [math.CO], 2018.

Masato Kobayashi, Integral representations for zeta(3) with the inverse sine function, arXiv:2108.01247 [math.NT], 2021.

Richard J. Mathar, Table of Dirichlet L-series and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 2010-2015, section 2.2 entry L(m=4,r=2,s=3).

Eric Weisstein's World of Mathematics, Dirichlet Beta Function.

Wikipedia, Dirichlet beta function.

Formula

chi4(k) = Kronecker(-4, k); chi4(k) is 0, 1, 0, -1 when k reduced modulo 4 is 0, 1, 2, 3, respectively; chi4 is A101455.

Series: L(3, chi4) = Sum_{k>=1} chi4(k) k^{-3} = 1 - 1/3^3 + 1/5^3 - 1/7^3 + 1/9^3 - 1/11^3 + 1/13^3 - 1/15^3 + ...

Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) Pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]

Closed form: L(3, chi4) = Pi^3/32.

Equals Sum_{n>=0} (-1)^n/(2*n+1)^3. - Jean-François Alcover, Mar 29 2013

Equals Product_{k>=3} (1 - tan(Pi/2^k)^4) (Groenman, 1990). - Amiram Eldar, Apr 03 2022

Equals Integral_{x=0..1} arcsinh(x)*arccos(x)/x dx (Kobayashi, 2021). - Amiram Eldar, Jun 23 2023

From Amiram Eldar, Nov 06 2023: (Start)

Equals beta(3), where beta is the Dirichlet beta function.

Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^3)^(-1). (End)

Example

L(3, chi4) = Pi^3/32 = 0.9689461462593693804836348458469186...

Mathematica

nmax = 1000; First[ RealDigits[Pi^3/32, 10, nmax] ]

Prog

(PARI) Pi^3/32 \\ Michel Marcus, Aug 15 2018

Crossrefs

Cf. A101455, A153072, A153073, A153074.

Cf. A233091, A251809. [Bruno Berselli, Dec 10 2014]

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A175572 (beta(4)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Sequence in context: A138500 A161484 A103985 * A336085 A363539 A086279

Adjacent sequences: A153068 A153069 A153070 * A153072 A153073 A153074

Keyword

nonn,cons,easy

Author

Stuart Clary, Dec 17 2008

Extensions

Offset corrected by R. J. Mathar, Feb 05 2009

Status

approved