Derivatives of $L$-series of weakly holomorphic cusp forms

TL;DR

This paper derives explicit integral formulas for the derivatives of $L$-series associated with weakly holomorphic cusp forms, improving computational methods and providing explicit examples.

Contribution

It introduces new explicit formulas for derivatives of $L$-series of weakly holomorphic cusp forms, enhancing computational techniques.

Findings

Explicit formulas for derivatives of $L$-series as integrals within the upper half-plane.

Computational advantages demonstrated for classical holomorphic cusp forms.

Discussion of explicit examples and computational aspects.

Abstract

Based on the theory of $L$-series associated with weakly holomorphic modular forms in \cite{DLRR}, we derive explicit formulas for central values of derivatives of $L$-series as integrals with limits inside the upper half-plane. This has computational advantages, already in the case of classical holomorphic cusp forms and, in the last section, we discuss computational aspects and explicit examples.