$L$-values for conductor $32$

TL;DR

This paper extends methods to express elliptic curve L-values as periods by analyzing Eisenstein series and their Fourier expansions, providing explicit period representations and simple integral formulas for generating functions of these L-values.

Contribution

It develops a framework to write L-values of elliptic curves with conductor 32 as explicit periods using Eisenstein series and their Fourier expansions.

Findings

Expressed L-values as periods via Eisenstein series

Derived explicit integral formulas for L-values at specific points

Provided simple formulas for generating functions of L(E,k)

Abstract

In recent years, Rogers and Zudilin developed a method to write $L$-values attached to elliptic curves as periods. In order to apply this method to a broader collection of $L$-values, we study Eisenstein series and determine their Fourier series at cusps. Subsequently, we write the $L$-values of an elliptic curve of conductor 32 as an integral of Eisenstein series and evaluate the value at $k>1$ explicitly as a period. As a side result, we give simple integral expressions for the generating functions of $L(E,k)$ when even (or odd) $k$ runs over positive integers.