Luo, Rudnick, and Sarnak prove that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$, then

$|\alpha_{j,\pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{n^2+1}}$

for all $1\leq j\leq n$ and any unramified prime $p$. The ramification restriction was removed by Blomer and Brumley.

For $1\leq k\leq K$, let $\pi_k$ be a cuspidal automorphic representation of $\mathrm{GL}_{n_k}(\mathbb{A}_{\mathbb{Q}})$. Consider the isobaric sum $\Pi = \pi_1\boxplus \pi_2 \boxplus \cdots\boxplus \pi_K$. Then

$L(s,\Pi) = \prod_{k=1}^K L(s,\pi_k)$,

and a bound for the local roots of $\Pi$ at $p$ reduces to a combination of the bounds for each of the cuspidal constituents $\pi_k$. Here is a convenient and uniform (yet sub-optimal) bound: If $N$ is the maximum of the $n_k$'s, then

$|\alpha_{j,\Pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{N^2+1}}$, $1\leq j\leq n_1+\cdots+n_K$.

If $\pi$ (resp. $\pi'$) is a cuspidal automorphic representation of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$ (resp. $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$), then one has

$|\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{\frac{1}{2}-\frac{1}{m^2+1}+\frac{1}{2}-\frac{1}{n^2+1}}=p^{1-\frac{1}{m^2+1}-\frac{1}{n^2+1}}$ ($1\leq j\leq m$ and $1\leq j'\leq n$)

This follows from the explicit description of the local roots (including at the ramified primes) given by Brumley in the appendix to this paper.

ADDED: In the Selberg class, if you strictly adhere to the axioms, there is no notion of $\alpha_{j,\pi}(p)$. One has a Dirichlet series for $L(s)$ and for $\log L(s)$, but you are not guaranteed that the Dirichlet series for $L(s)$ factors as a product of local roots like

$\displaystyle \prod_{p}\prod_{j=1}^m(1-\alpha_{j,\pi}(p)p^{-s})^{-1}$.

Luo, Rudnick, and Sarnak prove that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$, then $$ |\alpha_{j,\pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{n^2+1}} $$ for all $1\leq j\leq n$ and any unramified prime $p$. The ramification restriction was removed by Blomer and Brumley.

For $1\leq k\leq K$, let $\pi_k$ be a cuspidal automorphic representation of $\mathrm{GL}_{n_k}(\mathbb{A}_{\mathbb{Q}})$. Consider the isobaric sum $\Pi = \pi_1\boxplus \pi_2 \boxplus \cdots\boxplus \pi_K$. Then $$ L(s,\Pi) = \prod_{k=1}^K L(s,\pi_k), $$ and a bound for the local roots of $\Pi$ at $p$ reduces to a combination of the bounds for each of the cuspidal constituents $\pi_k$. Here is a convenient and uniform (yet sub-optimal) bound: If $N$ is the maximum of the $n_k$'s, then $$ |\alpha_{j,\Pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{N^2+1}},\quad 1\leq j\leq n_1+\cdots+n_K. $$ If $\pi$ (resp. $\pi'$) is a cuspidal automorphic representation of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$ (resp. $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$), then one has $$ |\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{\frac{1}{2}-\frac{1}{m^2+1}+\frac{1}{2}-\frac{1}{n^2+1}}=p^{1-\frac{1}{m^2+1}-\frac{1}{n^2+1}} $$ where $1\leq j\leq m$ and $1\leq j'\leq n$.

This follows from the explicit description of the local roots (including at the ramified primes) given by Brumley in the appendix to this paper.

ADDED: In the Selberg class, if you strictly adhere to the axioms, there is no notion of $\alpha_{j,\pi}(p)$. One has a Dirichlet series for $L(s)$ and for $\log L(s)$, but you are not guaranteed that the Dirichlet series for $L(s)$ factors as a product of local roots like $$ \displaystyle \prod_{p}\prod_{j=1}^m(1-\alpha_{j,\pi}(p)p^{-s})^{-1}. $$

Luo, Rudnick, and Sarnak prove that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$, then

$|\alpha_{j,\pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{n^2+1}}$

for all $1\leq j\leq n$ and any unramified prime $p$. The ramification restriction was removed by Blomer and Brumley.

For $1\leq k\leq K$, let $\pi_k$ be a cuspidal automorphic representation of $\mathrm{GL}_{n_k}(\mathbb{A}_{\mathbb{Q}})$. Consider the isobaric sum $\Pi = \pi_1\boxplus \pi_2 \boxplus \cdots\boxplus \pi_K$. Then

$L(s,\Pi) = \prod_{k=1}^K L(s,\pi_k)$,

and a bound for the local roots of $\Pi$ at $p$ reduces to a combination of the bounds for each of the cuspidal constituents $\pi_k$. Here is a convenient and uniform (yet sub-optimal) bound: If $N$ is the maximum of the $n_k$'s, then

$|\alpha_{j,\Pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{N^2+1}}$, $1\leq j\leq n_1+\cdots+n_K$.

If $\pi$ (resp. $\pi'$) is a cuspidal automorphic representation of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$ (resp. $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$), then one has

$|\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{\frac{1}{2}-\frac{1}{m^2+1}+\frac{1}{2}-\frac{1}{n^2+1}}=p^{1-\frac{1}{m^2+1}-\frac{1}{n^2+1}}$ ($1\leq j\leq m$ and $1\leq j'\leq n$)

This follows from the explicit description of the local roots (including at the ramified primes) given by Brumley in the appendix to this paper.

Luo, Rudnick, and Sarnak prove that if $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_n(\mathbb{A}_{\mathbb{Q}})$, then

$|\alpha_{j,\pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{n^2+1}}$

for all $1\leq j\leq n$ and any unramified prime $p$. The ramification restriction was removed by Blomer and Brumley.

For $1\leq k\leq K$, let $\pi_k$ be a cuspidal automorphic representation of $\mathrm{GL}_{n_k}(\mathbb{A}_{\mathbb{Q}})$. Consider the isobaric sum $\Pi = \pi_1\boxplus \pi_2 \boxplus \cdots\boxplus \pi_K$. Then

$L(s,\Pi) = \prod_{k=1}^K L(s,\pi_k)$,

and a bound for the local roots of $\Pi$ at $p$ reduces to a combination of the bounds for each of the cuspidal constituents $\pi_k$. Here is a convenient and uniform (yet sub-optimal) bound: If $N$ is the maximum of the $n_k$'s, then

$|\alpha_{j,\Pi}(p)|\leq p^{\frac{1}{2}-\frac{1}{N^2+1}}$, $1\leq j\leq n_1+\cdots+n_K$.

If $\pi$ (resp. $\pi'$) is a cuspidal automorphic representation of $\mathrm{GL}_m(\mathbb{A}_{\mathbb{Q}})$ (resp. $\mathrm{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$), then one has

$|\alpha_{j,j',\pi\times\pi'}(p)|\leq p^{\frac{1}{2}-\frac{1}{m^2+1}+\frac{1}{2}-\frac{1}{n^2+1}}=p^{1-\frac{1}{m^2+1}-\frac{1}{n^2+1}}$ ($1\leq j\leq m$ and $1\leq j'\leq n$)

This follows from the explicit description of the local roots (including at the ramified primes) given by Brumley in the appendix to this paper.

ADDED: In the Selberg class, if you strictly adhere to the axioms, there is no notion of $\alpha_{j,\pi}(p)$. One has a Dirichlet series for $L(s)$ and for $\log L(s)$, but you are not guaranteed that the Dirichlet series for $L(s)$ factors as a product of local roots like

$\displaystyle \prod_{p}\prod_{j=1}^m(1-\alpha_{j,\pi}(p)p^{-s})^{-1}$.