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The Hill operator on the real line, is self-adjoint and its spectrum has a band-gap structure, that is, the intervals of continuous spectrum alternate with spectral gaps. The endpoints of these gaps are eigenvalues of the same differential operator L but considered on the interval [0,\pi] with periodic or antiperiodic boundary conditions. Considering the Hill operator in case of the specific potential v(x)=2acos(2x), so called the Mathieu operator, we provide precise asymptotics of the spectral gaps of L. Our result extends the asymptotic formula of Harrell--Avron--Simon by providing more asymptotic terms. This is joint work with Plamen Djakov.