A special value of the spectral zeta function of the non-commutative harmonic oscillators

The non-commutative harmonic oscillator is a 2×2-system of harmonic oscillators with a non-trivial correlation. We write down explicitly the special value at s=2 of the spectral zeta function of the non-commutative harmonic oscillator in terms of the complete elliptic integral of the first kind, which is a special case of a hypergeometric function. The research of the author is supported in part by a Grant-in-Aid for Scientific Research (B) 15340005 from the Ministry of Education, Culture, Sports, Science and Technology.