  Modular forms on the split exceptional group $G_2$ over $\mathbb{Q}$ are a
special class of automorphic forms on this group, which were introduced by Gan,
Gross, and Savin. If $\pi$ is a cuspidal automorphic representation of
$G_2(\mathbb{A})$ corresponding to a level one, even weight modular form
$\varphi$ on $G_2$, we define an associated completed standard $L$-function,
$\Lambda(\pi,Std,s)$. Assuming that a certain Fourier coefficient of $\varphi$
is nonzero, we prove the functional equation $\Lambda(\pi,Std,s) =
\Lambda(\pi,Std,1-s)$. The proof proceeds via a careful analysis of a
Rankin-Selberg integral due to Gurevich and Segal.
