For $GL(3)$, the exterior square $L$-function $L(s,\wedge^2\pi)$ is entire as it agrees with $L(s,\tilde\pi\otimes\omega)$, where $\omega$ is the central character of $\pi$. Therefore, $L(1,\mathrm{sym}^2\pi)=0$ would imply that $L(1,\pi\otimes\pi)=0$, contradicting a result of [Shahidi (1980)][1]. The best known zero-free region for general Rankin-Selberg $L$-functions is due to Brumley (2012): see the Appendix [here][2].


  [1]: https://projecteuclid.org/euclid.bams/1183546358
  [2]: https://www.esi.ac.at/static/esiprpr/esi2403.pdf