nt.number theory - Definition of L-function attached to automorphic representation

As far as I understand, attaching an $L$-function to an automorphic representation attached to a general reductive group $G$ is conjectural and still open.

The way one attaches $L$-function depends on a representation $r$ of ${^L}G$ and partitioning the set of irreducible admissible representation of $G(k_v)$ in to $L$-packets (which is conjectural in general and known in very few cases). Assuming one can define local $L$-packets, Borel and Tate's article in Corvalis explains how to attach $L$-function to it. But still this $L$-function depends on the chosen representation $r$.

If $pi$ is an irreducible admissible representation of $G_A$ then $pi= otimes_v pi_v$, where $pi_v$ is an irreducible admissible representation of $G(k_v)$. So assuming we can partition the set of irreducible admissible representation of $G(k_v)$ in to $L$-packets, $pi_v$ belongs to $L$-packet $Pi_{phi_v}$ corresponding to some admissible homomorphism $phi_v$ of Weil-Deligne group to ${^L}G/k_v$ . The repesenation $r$ defines a representation $r_v$ of ${^L}G/k_v$.

Then the $L$-function attached to $pi$ and $r$ is defined as:
$L(s,pi,r) = prod_v L(s,pi_v,r_v)$,
$L(s,pi_v,r_v)=L(s, r_v circ phi_v)$

Now $r_v circ phi_v$ is a represenatation Wiel-Deilgne group, so by Tate's article in Corvalis, we know local $L$-factor.

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