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A153071 Decimal expansion of L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal 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 (list; constant; graph; refs; listen; history; text; internal format)
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.

R. J. Mathar, Table of Dirichlet L-series 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 [Jean-Fran├žois 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

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Last modified November 23 09:20 EST 2014. Contains 249840 sequences.