Suppose $\pi$ is an irreducible automorphic representation of a reductive connected algebraic group $G$ over $\mathbb{A}_K$, here $K$ is a number field, $\mathbb{A}_K$ denotes its adeles. We have a restricted tensor product decomposition of $\pi=\otimes\pi_v$, where $\pi_v$ is an irreducible admissible representation for $G(K_v)$, and for all but finitely many $v$, $\pi_v$ is unramified.

We know how to define local L-factors at $v$ is $\pi_v$ is unramified, and we also know how to define local L-factors at archimedean places because of Langlands classification. So the question is how to define L-factors at ramified places?

As far as I know, at least for $GL_n$, we can define it as the gcd of some family of integrals via integral representation of L-function.