  In the mirror symmetry of Calabi-Yau threefolds, the instanton expansion of
the prepotential has a constant term that is a rational multiple of
$\zeta(3)/(2 \pi i)^3$, the motivic origin of which has been carefully studied
in the author's paper with M. Kim. The Gamma conjecture claims that higher zeta
values in fact appear in the theory of Calabi-Yau $n$-folds for $n \geq 4$.
Moreover, a natural question is whether they also have a motivic origin. In
this paper, we will study the limit mixed Hodge structure (MHS) of the Fermat
pencil of Calabi-Yau $n$-folds at the large complex structure limit, which is
actually a mixed Hodge-Tate structure. We will compute the period matrix of
this limit MHS for the cases where $n=4,5,6,7,8,9,10,11,12$ by numerical
method, and our computations have shown the occurrence of higher zeta values,
which provide evidence to the Gamma conjecture. Furthermore, we will also
provide a motivic explanation to our numerical results.
