  The goal of this paper is to provide a complete and refined study of the
standard $L$-functions $L(\pi,\operatorname{Std},s)$ 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,
$\Lambda(\pi,\operatorname{Std},s)$. Assuming that a certain Fourier
coefficient of $\varphi$ is nonzero, we prove the functional equation
$\Lambda(\pi,\operatorname{Std},s) = \Lambda(\pi,\operatorname{Std},1-s)$. Our
proof proceeds via a careful analysis of a Rankin-Selberg integral that is due
to an earlier work of Gurevich and Segal.
