A003420 Values of m in the discriminant D = -4*m leading to a new maximum of the L-function of the Dirichlet series L(1) = Sum_{k=1..oo} Kronecker(D,k)/k.

1, 2, 5, 11, 14, 26, 41, 89, 101, 194, 314, 341, 689, 1091, 1154, 1889, 2141, 3449, 3506, 5561, 6254, 8126, 8774, 10709, 13166, 15461, 23201, 24569, 30014, 81149, 81626, 162686, 243374, 644474, 839354, 879941

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• more

Comments

In Shanks's Table 5 "Hichamps, -4N = Discriminant", N = 1 is omitted, and N = 23201 is missing. Shanks describes the table as being tentative after N = 24569. In Buell's Table 10 "Successive maxima of L(1) for even discriminants", the values N = 11 and N = 1091 are missing in the D/4 column. The further terms 644474, 839354, 879941, provided there require an independent check. - Hugo Pfoertner, Feb 02 2020

Examples

			a(1) = 1: L(1) for D=-4*1 ~= 0.785398... = Pi/4.
a(2) = 2: L(1) for D=-4*2 ~= 1.11072073... = Pi/(2*sqrt(2)), a(2) > a(1);
L(1) for D=-4*3 ~= 0.90689..., L(1) for D=-4*4 ~= 0.785398..., both < a(2);
a(3) = 5: L(1) for D=-4*5 = 1.40496..., a(3) > a(2).
		

References

• D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), pp. 267-283 of Analytic Number Theory, ed. H. G. Diamond, Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• Duncan A. Buell, Small class numbers and extreme values of L-functions of quadratic fields, Math. Comp., 31 (1977), 786-796 (Table 10, page 792).
• D. Shanks, Systematic examination of Littlewood's bounds on L(1,chi), Proc. Sympos. Pure Math., 24 (1973). Amer. Math. Soc. (Annotated scanned copy)

Crossrefs

Cf. A003521.

Cf. A331949, which has almost identical terms.

Extensions

New title, a(1) prepended, missing term 23201 and a(29)-a(33) from Hugo Pfoertner, Feb 02 2020

3 further terms < 10^6 added by Hugo Pfoertner, Aug 27 2022