Let $G$ be a connected, reductive group over a $p$-adic field $k$. Let $\pi$ be an irreducible, admissible representation of $G(k)$, and $r$ a finite dimensional continuous representation of the $L$-group $^LG$ of $G$ whose restriction to $^LG^0$ is analytic. Conjecturally, there should be a local L-function $L(s,\pi,r)$ and $\epsilon$-factor $\epsilon(s,\pi,r,\psi)$ attached to $\pi$ and $r$ (for a chosen additive character $\psi$ of $k$), but so far there are no general definitions. They are defined for example if $G$ and $\pi$ are unramified.
What about when $G= T$ is a torus? Are there definitions for $L(s,\pi,r)$ and $\epsilon(s,\pi,r,\psi)$, for $\pi$ a continuous homomorphism $T(k) \rightarrow \mathbb C^{\ast}$?
