Mean values of L-funtions associated to elliptic, fermat and other curves at the centre of the critical strip

On the Vanishing of TwistedL-Functions of Elliptic Curves

“…Their conjectures agree with the known results for the first few integral moments of the Riemann zeta-function (see [12,13]), and with the known results the first few integral moments of twists by quadratic Dirichlet characters (see [10,15,26]). They also agree with the number theoretic heuristics of [3,4].…”

“…Moments of cubic twists for the eight selected elliptic curves. The empirical moments are the moments (10) for various values of s and up to X given in Table 1. The conjectural moments are computed following Conjecture 3.2 with the arithmetic factor a E (s) of Section 4.…”

Mean values of biquadratic zeta functions

“…Jutila [11], verifying a conjecture of Goldfeld and Viola [10], showed that (1.1) 0<±d<X L(1/2, χ d ) ∼ c 1 X log X + c 2 x + O(x 3/4+ ) for certain constants c 1 , c 2 . Takhtadzjan and Vinogradov [15] establish a similar result.…”

Mean value of the class number in function fields revisited

“…But we can also consider the problem of averaging over fundamental discriminants. In 1979, Goldfeld and Viola [10] put forward a few conjectures about averages over fundamental discriminants and Jutila [13] proved such conjecture in 1981 for s = 1 2 . Then, in 1985, Goldfeld and Hoffstein [9], using Eisentein series of 1 2 -integral weight, generalized Jutila's result and proved a result valid for all s with Re(s) ≥ 1 2 .…”