$L(s,\pi_1\times \pi_2)$ has a pole at $s=1$ if $\pi_2=\tilde\pi_1$, but it is not neccesarily simple.

I'm sure that $L(s,\pi\times \tilde\pi)/\zeta_F(s)$ is holomorphic for some $\pi$, but I suspect that the problem should be open in general. Note that we don't know if $\zeta_F(s)/\zeta(s)$ is an entire function for non-Galois $F$.

Warning. I was thinking on $\pi$ not neccesarily cuspidal. Jeremy Rouse's answer is the correct one.

$L(s,\pi_1\times \pi_2)$ has a pole at $s=1$ if $\pi_2=\tilde\pi_1$, but it is not neccesarily simple.

I'm sure that $L(s,\pi\times \tilde\pi)/\zeta_F(s)$ is holomorphic for some $\pi$, but I suspect that the problem should be open in general. Note that we don't know if $\zeta_F(s)/\zeta(s)$ is an entire function for non-Galois $F$.

$L(s,\pi_1\times \pi_2)$ has a pole at $s=1$ if $\pi_2=\tilde\pi_1$, but it is not neccesarily simple.

I'm sure that $L(s,\pi\times \tilde\pi)/\zeta_F(s)$ is holomorphic for some $\pi$, but I suspect that the problem should be open in general. Note that we don't know if $\zeta(s)/\zeta_F(s)$ is an entire function for non-Galois $F$.

$L(s,\pi_1\times \pi_2)$ has a pole at $s=1$ if $\pi_2=\tilde\pi_1$, but it is not neccesarily simple.

I'm sure that $L(s,\pi\times \tilde\pi)/\zeta_F(s)$ is holomorphic for some $\pi$, but I suspect that the problem should be open in general. Note that we don't know if $\zeta_F(s)/\zeta(s)$ is an entire function for non-Galois $F$.