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