Abstract: We investigate the spectral zeta function of the
non-commutative harmonic oscillator studied in Parmeggiani and Wakayama.
It is shown, as one of the basic analytic properties, that the spectral
zeta function is extended to a meromorphic function in the whole complex
plane with a simple pole at $s=1$, and further that it has a zero at all
non-positive even integers, i.e. at $s=0$ and at those negative even
integers where the Riemann zeta function has the so-called trivial zeros.
As a by-product of the study, both the upper and the lower bounds
are also given for the first eigenvalue of the non-commutative harmonic
oscillator. It is a common work with M. Wakayama.