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

Rogers, M.; Wan, J. G.; Zucker, I. J.

Ramanujan J. 37, No. 1, 113-130 (2015).

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.

MSC:

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

Keywords:

moments of elliptic integrals; critical \(L\)-values; hypergeometric functions; Gamma functions; lattice sums

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