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The problem of non-existance of Siegel-Landau zeros seems to be uncharacteristically easier for cuspidal automorphic representations $\pi$ on $\mathrm{GL}(n)$ if $n\geq2$. We have in fact:

There are no exceptional zeros on $\mathrm{GL}(2)$

This was proved for all non-dihedral cusp forms over arbitrary number fields by Ramahrishnan and Hoffstein (1995). Also,

There are no exceptional zeros on $\mathrm{GL}(3)$

This is a theorem of Banks (1997). I'm also aware that the result for $\mathrm{GL}(n)$ follows from Langlands functoriality (only for $n>1$, of course; for $n=1$ the problem seems to lie even deeper).

There's also a wealth of results for symmetric and adjoint lifts, Rankin-Selberg L-functions...

My question is, what is the situation for $n\geq 4$?

As GH mentions in the comments, the absence of exceptional zeros was also proved by Ramahrishnan and Hoffstein for all non-selfdual cusp forms on $\mathrm{GL}(n)$ over number fields.

No other unconditional progress seems to have been made, or at least I can't find any mention of it.

The results above were inspired by a lecture by Dorian Goldfeld in October 1994. Given how quickly (and relatively easy) progress was made in the first cases, is there any new exceptional behaviour in $\mathrm{GL}(4)$, or we do just simply run out of technical tools?

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    $\begingroup$ Your question is very good, but there is some confusion in your post. First, a Maass form is also a cusp form. Second, Hoffstein-Lockhart proved a Siegel type ineffective lower bound for $L(1,\mathrm{ad}^2\pi)$, but they did not prove the absence of Siegel zeros. Third, Hoffstein-Ramakrishnan proved the absence of Siegel zeros for all non-dihedral $\mathrm{GL}_2$ cusp forms, even over number fields. Note that for dihedral forms the problem is equivalent to the case of $\mathrm{GL}_1$, which is unresolved. $\endgroup$
    – GH from MO
    Nov 18, 2015 at 1:33
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    $\begingroup$ Continued: Finally, it should also be mentioned that Hoffstein-Ramakrishnan proved the absence of Siegel zeros for all non-selfdual cusp forms on $\mathrm{GL}_n$ over a number field. $\endgroup$
    – GH from MO
    Nov 18, 2015 at 1:37

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