  The purpose of the present paper is to study the number theoretic properties
of the special values of the spectral zeta functions of the non-commutative
harmonic oscillator (NcHO), especially in relation to modular forms and
elliptic curves from the viewpoint of Fuchsian differential equations, and
deepen the understanding of the spectrum of the NcHO. We study first the
general expression of special values of the spectral zeta function $\zeta_Q(s)$
of the NcHO at $s=n$ $(n=2,3,\dots)$ and then the generating and
meta-generating functions for Ap\'ery-like numbers defined through the analysis
of special values $\zeta_Q(n)$. Actually, we show that the generating function
$w_{2n}$ of such Ap\'ery-like numbers appearing (as the "first anomaly") in
$\zeta_Q(2n)$ for $n=2$ gives an example of automorphic integral with rational
period functions in the sense of Knopp, but still a better explanation remains
to be clarified explicitly for $n>2$. This is a generalization of our earlier
result on showing that $w_2$ is interpreted as a $\Gamma(2)$-modular form of
weight $1$. Moreover, certain congruence relations over primes for "normalized"
Ap\'ery-like numbers are also proven. In order to describe $w_{2n}$ in a
similar manner as $w_2$, we introduce a differential Eisenstein series by using
analytic continuation of a classical generalized Eisenstein series due to
Berndt. The differential Eisenstein series is actually a typical example of the
automorphic integral of negative weight. We then have an explicit expression of
$w_4$ in terms of the differential Eisenstein series. We discuss also shortly
the Hecke operators acting on such automorphic integrals and relating Eichler's
cohomology group.
