  We discuss odd-frequency Cooper pairs appearing in two-band superconductors
by solving the Gor'kov equation analytically. We introduce the equal-time
$s$-wave pair potentials as realized in MgB$_2$ and iron pnictides. Although
the order parameter symmetry is conventional, the band degree of freedom
enriches the symmetry variety of pairing correlations. The hybridization and
the asymmetry between the two conduction bands induce odd-frequency pairs as a
subdominant pairing correlation in the uniform ground state. To study the
magnetic response of odd-frequency Cooper pairs, we analyze the Meissner kernel
represented by the Gor'kov Green function. In contrast to the even-frequency
pairs linked to the pair potential, the induced odd-frequency Cooper pairs
indicate a paramagnetic property. We also discuss the relation between the
amplitude of the odd-frequency pairing correlation and the stability of
superconducting states in terms of the self-consistent equation for the pair
potential.
