Let $\pi$ be an automorphic representation of $GL(2)/\mathbb{Q}$. For simplicity, you can take it to be a Maass form for $SL(2,\mathbb Z)$. Kim, Shahidi, Gelbart-Jacquet prove that $$L(s, \pi, Sym^m)$$ is automorphic on $GL(m+1)$ for $m=2,3,4$. And therefore the analytic property (being entire) of $L(s, \pi, Sym^m)$ is known for $m=2,3,4$.

What is known for $m=5$ or higher? I think we know that for $m=5,6,7,8$, $L(s, \pi, Sym^m)$ is meromorphic and satisfies functional equation. Do we know that it is holomorphic on any (small) area on $\{1-\delta<\Re s\leq 1\}$ for some small $\delta$? Kim-Shahidi's DUKE paper can do it for $\Re s\geq 1$.

PS: Since Langlands-Shiahidi method could yield $Sym^4 \pi\times \pi'$ with $\pi'$ on GL(4), I can't believe the same method fails to say anything about $Sym^5\pi$.