Computing L-functions of $ λ$-adic representations of global function fields

Abstract

The L-function $ L(ρ_λ, s) $ of an almost everywhere unramified $ λ$-adic representation $ ρ_λ$ of a global function field $ \mathbb{F}_q(C) $ is known to be a rational function in $ q^{-s} $ satisfying a functional equation up to some complex sign $ ε(ρ_λ) $. This paper presents a systematic framework to compute the coefficients of $ L(ρ_λ, s) $ and its sign $ ε(ρ_λ) $ with some explicit examples.