

A153071


Decimal expansion of L(3, chi4), where L(s, chi4) is the Dirichlet Lfunction for the nonprincipal character modulo 4.


21



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
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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, SpringerVerlag, 1989. See page 293, Entry 25 (iii).


LINKS

Table of n, a(n) for n=0..89.
QingHu Hou, ZhiWei Sun, A qanalogue of the identity Sum_{k>=0}(1)^k/(2k+1)^3 = Pi^3/32, arXiv:1808.04717 [math.CO], 2018.
R. J. Mathar, Table of Dirichlet Lseries and prime zeta modulo functions for small moduli, arXiv:1008.2547 [math.NT], 20102015, section 2.2 entry L(m=4,r=2,s=3).


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. [JeanFrançois Alcover, Mar 29 2013]


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. A153072, A153073, A153074, A175570, A175571, A175572.
Cf. A233091, A251809. [Bruno Berselli, Dec 10 2014]
Sequence in context: A138500 A161484 A103985 * A086279 A155533 A083281
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



