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The identity $L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\tau_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, Proc. Symp. Pure Math. 33 (19771979), Part 2, 63-86.
The identity $L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\tau_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, Proc. Symp. Pure Math. 33 (1977), 63-86.
The identity $L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\tau_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, Proc. Symp. Pure Math. 33 (1979), Part 2, 63-86.
I am not sure if I understand your questionThe identity $L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\tau_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, but let me try to answerProc. Symp. Pure Math. 33 (1977), 63-86.
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.
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
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 $\prod_i L(\tau_i,s)=\prod_i L(\rho'_i,s)$.
I am not sure if I understand your question, but let me try to answer.
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.
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
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 $\prod_i L(\tau_i,s)=\prod_i L(\rho'_i,s)$.
The identity $L(Q(\tau_1,\ldots,\, \tau_r),\,s) = \prod_i L(\tau_i,\,s)$ is part of Theorem 3.4 in Jacquet: Principal $L$-functions of the linear group, Proc. Symp. Pure Math. 33 (1977), 63-86.
I am not sure if I understand your question, but let me try to answer.
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.
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
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 $\prod_i L(\tau_i,s)=\prod_i L(\rho'_i,s)$.
