Computing special values of motivic L-functions

TL;DR

This paper introduces an algorithm for numerically computing special values and derivatives of motivic L-functions, accommodating complex Gamma-factors and relying on their functional equations.

Contribution

It provides a novel numerical method applicable to a broad class of motivic L-series with complex Gamma-factors, expanding computational capabilities.

Findings

Algorithm accurately computes L-values and derivatives

Applicable to L-series with complex Gamma-factors

Relies on known or conjectural functional equations

Abstract

We present an algorithm to compute values L(s) and derivatives of L-functions of motivic origin numerically to required accuracy. Specifically, the method applies to any L-series whose Gamma-factor is a product of any number of Gamma-functions Gamma((s+l(j))/2) with complex l(j), not necessarily distinct. The algorithm relies on the known (or conjectural) functional equation for L(s).