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.