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

%I A153072 #10 Jun 02 2025 01:15:51
%S A153072 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,
%T A153072 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,
%U A153072 20,2,1,5,10,1,4,2,2,1,2,106,4,1,1,1,1,1,10,9,3,3,14
%N A153072 Continued fraction for L(3, chi4), where L(s, chi4) is the Dirichlet L-function for the non-principal character modulo 4.
%D A153072 Leonhard Euler, "Introductio in Analysin Infinitorum", First Part, Articles 175, 284 and 287.
%D A153072 Bruce C. Berndt, "Ramanujan's Notebooks, Part II", Springer-Verlag, 1989. See page 293, Entry 25 (iii).
%F A153072 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.
%F A153072 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 + ...
%F A153072 Series: L(3, chi4) = Sum_{k>=0} tanh((2k+1) Pi/2)/(2k+1)^3. [Ramanujan; see Berndt, page 293]
%F A153072 Closed form: L(3, chi4) = Pi^3/32.
%e A153072 L(3, chi4) = 0.9689461462593693804836348458469186... = [0; 1, 31, 4, 1, 18, 21, 1, 1, 2, 1, 2, 1, 3, 6, 3, 28, ...].
%t A153072 nmax = 1000; ContinuedFraction[Pi^3/32, nmax + 1]
%Y A153072 Cf. A153071, A153073, A153074.
%K A153072 nonn,cofr,easy
%O A153072 0,3
%A A153072 _Stuart Clary_, Dec 17 2008