Hypergeometric expressions of $L$-values for a Borweins theta product of weight $3$

TL;DR

This paper expresses the special $L$-values of a weight 3 modular form, formed by Borweins theta products, in terms of Kampé de Fériet hypergeometric functions, linking modular forms and hypergeometric functions.

Contribution

It introduces a novel expression of $L$-values for a specific modular form using Kampé de Fériet hypergeometric functions, expanding the analytical tools for studying $L$-values.

Findings

$L$-values at $s=1,2,3$ are expressed via hypergeometric functions

Establishes a connection between modular forms and Kampé de Fériet functions

Provides explicit formulas for $L$-values of a weight 3 form

Abstract

In this paper, we consider a modular form of weight 3, which is a product of the Borweins theta series, and express its $L$-values at $s=1$, $2$ and $3$ in terms of special values of Kamp\'e de F\'eriet hypergeometric functions, which are two-variable generalization of generalized hypergeometric functions.