For simplicity, let $F$ be a number field, and assume that $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_{n}(\mathbb{A}_F)$ with trivial conductor. Miller and Schmid proved that $L(s,\pi;\bigwedge^2)$ is holomorphic on $\mathbb{C}-\{0,1\}$. Takeda proved that $L(s,\pi;\mathrm{Sym}^2)$ is holomorphic on $\mathrm{Re}(s)>1-\frac{1}{n}$ except for a possible pole at $s=1$. (There might be other references to establish the holomorphy in the half-plane $\mathrm{Re}(s)>1$, but these sources came to mind first.) We have the factorization
$L(s,\pi\times\pi) = L(s,\pi;\mathrm{Sym}^2) L(s,\pi;\bigwedge^2).$
Jacquet, Piatetski-Shapiro, and Shalika proved (among many other things) that $L(s,\pi\times\pi)$ is holomorphic on $\mathbb{C}-\{0,1\}$. They also proved that the Euler product that defines $L(s,\pi\times\pi)$ converges absolutely for $\mathrm{Re}(s)>1$. It follows that $L(s,\pi\times\pi)$ does not vanish in this region. We now conclude that $L(s,\pi;\mathrm{Sym}^2)$ and $L(s,\pi;\bigwedge^2)$ are both holomorphic and non-vanishing for $\mathrm{Re}(s)>1$. More generally, in the region $\mathrm{Re}(s)>1-1/n$ (and away from a possible pole at $s=1$), $L(s,\pi;\mathrm{Sym}^2)$ and $L(s,\pi;\bigwedge^2)$ are non-vanishing in whatever zero-free region is available for $L(s,\pi\times\pi)$. For arbitrary $\pi$, the best such zero-free region is due to Brumley (see the appendix).
If the conductor of $\pi$ is nontrivial, then everything needs to first be expressed in terms of partial $L$-functions, omitting the Euler factors at the places $v$ at which $\pi_v$ ramifies. The holomorphy of these Euler factors can be deduced from bounds of Luo--Rudnick--Sarnak type, but for the ramified places, which they do not address. For variants of the Luo--Rudnick--Sarnak progress towards the generalized Ramanujan conjecture at all places (not just the unramified ones), see Müller and Speh (Section 3) or Blomer and Brumley.
(For future posts: "It is written is some paper..." is far too vague.)
