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Let $\pi$ be a Maass cusp form for SL($2,\mathbb Z$). Let $\omega$ be a primitive Dirichlet character.

Let us consider the $L-$ function $$L(s,Sym^5 \pi \times \omega)$$ or $L(s,Sym^6 \pi \times \omega)$ or $L(s,Sym^7 \pi \times \omega)$ or $L(s,Sym^8 \pi \times \omega)$.

Are they known to be holomorphic on the whole complex plane?

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    $\begingroup$ Surely not. These sorts of results are only recently established in the holomorphic case and the techniques used there (Galois representations) won't generalise (for now at least). $\endgroup$
    – eric
    Jul 26, 2015 at 19:34
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    $\begingroup$ Hi Eric, but Galois representations are not the only method to treat $L$-functions. How about Langlands-Shahidi method? Do they includes any one of these cases? BTW, I would like to see your reference for modular forms. $\endgroup$
    – 7-adic
    Jul 26, 2015 at 21:11
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    $\begingroup$ Symm^5 is way out of reach using any method other than the automorphy lifting theorem techniques. Look at recent papers by Clozel and Thorne for the holomorphic case. $\endgroup$
    – eric
    Jul 26, 2015 at 21:36
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    $\begingroup$ @7-adic: Nope -- $Sym^5$ and higher are out of range of Langlands-Shahidi. Basically getting up to $Sym^4$ requires some special cases of Levi subgroups in exceptional groups, and sadly the special cases run out. Garland has a long-term program to try to extend these methods to infinite-dimensional groups -- if one is allowed to use Kac-Moody groups, one could go further. But alas, the results in that direction are not nearly strong enough for $Sym^5$ L-functions as far as I know. $\endgroup$
    – Marty
    Jul 26, 2015 at 23:05
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    $\begingroup$ Following up on @Marty's comment: in the 80s, several people, including PiateskiShapiro-Rallis, myself, and others, tried to identify what sort of Rankin-Selberg situation could produce such "higher" L-functions... tentatively thinking in terms of "generalized groups"... but/and found that (at least it seemed at the time) there was no sane "generalized group" recipe that could produce a given L-function "at will". Maybe there has been progress, but it is already not so easy to make a reductive "group" with arbitrarily specified Coxeter group as "Weyl group"... Such obstacles. $\endgroup$ Jul 27, 2015 at 21:41

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This is definitely an open problem.

For a very recent reference of $Sym^m\pi$ being known only for $m \leq 4$ ($\pi$ arbitrary cuspidal), see:

As eric mentions in the comments, arithmetic techniques make some cases like holomorphic forms accesible, but those techniques are certenly missing in this case, $\pi$ a Maass form.

Also, more concretely for Maass forms, see this paragraph on a 2003 Sarnak paper (Spectra of Hyperbolic Surfaces):

This [Kim-Sarnak result towards the eigenvalue conjecture for Maass forms] is getting close to $1/4$, but it is also close to the limit of these methods. The functorial lifts $sym^3$ and $sym^4$ are based on the continuous spectrum (Eisenstein series) on exceptional groups including $E_8$. What can be done this way terminates with the finite list of exceptional groups.

This is closely related with what Marty and Garret said in their very interesting comments above.

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