Computing L-functions of $ \lambda $-adic representations of global function fields

The L-function $L(\rho_\lambda, s)$ of an almost everywhere unramified $\lambda$-adic representation $\rho_\lambda$ 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 $\epsilon(\rho_\lambda)$. This paper presents a systematic framework to compute the coefficients of $L(\rho_\lambda, s)$ and its sign $\epsilon(\rho_\lambda)$ with some explicit examples.