  In this paper we consider the oscillatory integrals on Lefschetz thimbles in
the Landau-Ginzburg model as the mirror of a toric Fano manifold. We show these
thimbles represent the same relative homology classes as the characteristic
cycles of the corresponding constructible sheaves under the equivalence of
\cite{GPS18-2}. Then the oscillatory integrals on such thimbles are the same as
the integrals on the characteristic cycles and relate to genus $0$
Gromov-Witten descendant potential for $X$, and this leads to a proof of Gamma
II conjecture for toric Fano manifolds.
