I'm reading Kudla's Article on the Local Langlands Conjecture for $p$-adic general linear groups, and specifically I'm trying to understand how the ideas of Bernstein-Zelevinski yield show that you only need to prove the LLC for supercuspidal representations (on the automorphic side) and irreducible representatons (on the Galois side).
The part I can't find a reference on is: getting the $L$-functions to match. In particular, if $\tau_1,\,\ldots\,\,\tau_r$ are essentially-square-integrable representations, then they correspond to indecomposable representations $\rho_1',\,\ldots,\,\rho_r'$ on the Galois side. Then the correspondence gives $$Q(\tau_1,\ldots,\, \tau_r) \mapsto \rho_1' \oplus\ldots\oplus \rho_r'$$ (here $Q$ is the Langlands quotient, i.e. the unique irreducible quotient of the representation parabolically induced from $\tau_1\otimes \ldots\otimes \tau_r$). and so for match of $L$-functions we need $$L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\rho_i',\,s) = \prod_i L(\tau_i,\,s).$$
The fact that $$L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\tau_i,\,s)$$ Is stated in Kudla's article, but is not proved, and a reference is not provided. I was wondering if anyone could point me in the right direction, or explain why this is true?
(I guess the same question goes for $\epsilon$ factors but these are much more mysterious to me).
EDIT: unnecessary question removed, then edited for clarity.