Lower order terms for the moments of symplectic and orthogonal families of $L$-functions

Ian P. Goulden, Duc Khiem Huynh, Rishikesh, Michael O. Rubinstein

Published 2012-03-21, updated 2012-06-15Version 3

We derive formulas for the terms in the conjectured asymptotic expansions of the moments, at the central point, of quadratic Dirichlet $L$-functions, $L(1/2,\chi_d)$, and also of the $L$-functions associated to quadratic twists of an elliptic curve over $\Q$. In so doing, we are led to study determinants of binomial coefficients of the form $\det (\binom{2k-i-\lambda_{k-i+1}}{2k-2j})$.