Evaluating $L$-functions with few known coefficients

TL;DR

This paper presents a novel method for evaluating $L$-functions with limited known coefficients, achieving higher precision than standard methods at the cost of increased computational effort.

Contribution

The authors introduce a new approach using the approximate functional equation to evaluate $L$-functions more accurately with few known coefficients.

Findings

More precise evaluation than standard methods

Requires significantly more computation

Effective with limited Dirichlet coefficients

Abstract

We address the problem of evaluating an $L$-function when only a small number of its Dirichlet coefficients are known. We use the approximate functional equation in a new way and find that is possible to evaluate the $L$-function more precisely than one would expect from the standard approach. The method, however, requires considerably more computational effort to achieve a given accuracy than would be needed if more Dirichlet coefficients were available.