Moments of elliptic integrals and critical \(L\)-values. (English) Zbl 1383.11048

Summary: 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 we also obtain new lattice sum evaluations.


11F03 Modular and automorphic functions
33C20 Generalized hypergeometric series, \({}_pF_q\)
11M41 Other Dirichlet series and zeta functions
33C75 Elliptic integrals as hypergeometric functions
33E05 Elliptic functions and integrals
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