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.
Best Answer
I think you're slightly misled. $\pi$ doesn't have an $L$-function "in abstracta". An unramified $\pi_v$ at a finite place $v$ gives rise, by Langlands' interpretation of the Satake isomorphism, to a semi-simple conjugacy class in the local $L$-group (or, more fancily, to an unramified representation of the local Weil[-Deligne] group [with N=0]), and that's not enough for an Euler factor. So what one does is also fixes a representation $r:{}^LG\to GL_n(\mathbf{C})$. Now one has an $L$-function $L(\pi,r,s)$, it depends on both $\pi$ and $r$ though. For example, if you choose a modular elliptic curve over the rationals, but let $r$ be the bazillion'th tensor power of the standard 2-dimensional representation of $GL_2$, you have an $L$-function which no-one knows how to analytically continue.
But the real answer to your question is that this is an open problem. One would like to say that by functoriality there is a representation $r_*(\pi)$ on $GL_n$ and you use standard definitions of $L$-functions on $GL_n$. However the existence of $r_*(\pi)$ is a fundamental open problem: Langlands functoriality. Defining local $L$-functions at the ramified places is a tiny tiny piece of this open problem, but as far as I know it's also open.