Transformations of $L$-values

TL;DR

This paper introduces a new analytical method to express $L$-values of elliptic curves as periods, extending previous work on Mahler measures and Boyd's conjectures, with a focus on general settings and illustrative examples.

Contribution

It presents a novel analytical machinery for representing $L$-values as periods, broadening the scope of prior approaches and providing a simple illustrative example.

Findings

New method expresses $L(E,2)$ as periods.

Extended the approach to more general settings.

Provided an illustrative example demonstrating the method.

Abstract

In our recent work with Mat Rogers on resolving some Boyd's conjectures on two-variate Mahler measures, a new analytical machinery was introduced to write the values $L(E,2)$ of $L$-series of elliptic curves as periods in the sense of Kontsevich and Zagier. Here we outline, in slightly more general settings, the novelty of our method with Rogers, and provide a simple illustrative example.