  The Bombieri-Dwork conjecture predicts that the differential equations
satisfied by $G$-functions come from geometry. In this paper, we will look at
special $G$-functions whose differential equations have a special singularity
with maximally unipotent monodromy. We will formulate a Gamma conjecture about
such $G$-functions, which has close connections with the mirror symmetry of
Calabi-Yau threefolds and the Gamma conjecture in algebraic geometry. We will
provide examples to support this conjecture, which involves numerical
computations using Mathematica programs.
