A153072 Continued fraction for L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.

0, 1, 31, 4, 1, 18, 21, 1, 1, 2, 1, 2, 1, 3, 6, 3, 28, 1, 3, 2, 1, 2, 21, 1, 1, 32, 1, 1, 1, 5, 3, 1, 2, 1, 27, 11, 1, 2, 1, 5, 1, 3, 4, 3, 1, 4, 1, 1, 2, 1, 9, 8, 1, 2, 2, 1, 14, 2, 1, 7, 2, 2, 1, 20, 2, 1, 5, 10, 1, 4, 2, 2, 1, 2, 106, 4, 1, 1, 1, 1, 1, 10, 9, 3, 3, 14

Offset: 0

Author

Stuart Clary, Dec 17 2008

Keywords

• nonn
• cofr
• easy

Examples

			L(3, chi4) = 0.9689461462593693804836348458469186... = [0; 1, 31, 4, 1, 18, 21, 1, 1, 2, 1, 2, 1, 3, 6, 3, 28, ...].
		

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).

Crossrefs

Cf. A153071, A153073, A153074.

Programs

Mathematica

nmax = 1000; ContinuedFraction[Pi^3/32, nmax + 1]

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.