Moments of elliptic integrals and critical $L$-values

Abstract

We compute the critical $L$-values of some weight 3, 4, or 5 modular forms, by transforming them into integrals of the complete elliptic integral $K$. In doing so, we prove closed form formulas for some moments of $K^{\prime 3}$. Many of our $L$-values can be expressed in terms of Gamma functions, and this also gives new lattice sum evaluations.