Return to Revisions

I am not sure if I understand your question, but let me try to answer.

1. If you parabolically induce a local automorphic representation from $\tau_1,\dots,\tau_r$, then the $L$-function of the resulting local automorphic representation equals $\prod_i L(\tau_i,s)$. See Theorem 15.7.1 in Goldfeld-Hundley: Automorphic Representations and $L$-Functions for the General Linear Group, Volume 2.

2. If you take the direct sum of local Galois representations $\rho'_1,\dots,\rho'_r$, then the $L$-function of the resulting local Galois representation equals $\prod_i L(\rho'_i,s)$. See (IV) in Section VIII.3 (on Page 221) of Cassels-Fröhlich: Algebraic Number Theory

3. So matching the so-called isobaric sum of $\tau_1,\dots,\tau_r$ with the direct sum of $\rho'_1,\dots,\rho'_r$, through $L$-functions, means the identity $\prod_i L(\tau_i,s)=\prod_i L(\rho'_i,s)$.