To an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ one can associate its L-function $s\mapsto L_{\pi}(s)$.

Is the map $\pi\mapsto L_{\pi}$ bijective?

To an automorphic representation $\pi$ of $\operatorname{GL}_{n}(\mathbb{A}_{\mathbb{Q}})$ one can associate its L-function $s\mapsto L_{\pi}(s)$.

Is the map $\pi\mapsto L_{\pi}$ bijective?

Edit March 26th, 2021: can such a bijection be seen as a fully faithful fonctor between Tannakian categories $\mathcal{C}$ whose objects are automorphic representations and $\mathcal{D}$ whose objects are L-functions suggesting every automorphism of a given object of the latter arises from an intertwinning operator? If so, can such an automorphism $\varphi$ be written as $\varphi_{\sigma}:L_{\pi}\mapsto\sigma\circ L_{\pi}\circ\sigma^{-1}=L_{\pi}\mapsto{L_{\sigma'\circ\pi\circ\sigma'^{-1}}}$ where $\sigma$ is a field automorphism of $\mathbb{C}$ and $\sigma'$ the corresponding intertwinning operator?