The completed standard L-function of modular forms on $$G_2$$

The goal of this paper is to provide a complete and refined study the standard L-functions $$L(\pi ,{\text {Std}},s)$$ for certain non-generic cuspidal automorphic representations $$\pi $$ $$G_2({\mathbb {A}})$$ . For representation that corresponds modular form $$\varphi level one even weight on $$G_2$$ , we explicitly define completed L-function, $$\Lambda (\pi Assuming Fourier coefficient nonzero, prove functional equation {Std}},s) = \Lambda {Std}},1-s)$$ Our proof proceeds via careful analysis Rankin-Selberg integral due an earlier work Gurevich Segal.