

A153071


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


7



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.
R. J. Mathar, Table of Dirichlet Lseries and prime zeta modulo functions for small moduli, arXiv:1008.2547, 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..infinity} 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..infinity} 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) = 0.9689461462593693804836348458469186...


MATHEMATICA

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


CROSSREFS

Cf. A153072, A153073, A153074, A175570  A175572.
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



