  The subthreshold 1$^-_1$ state at an excitation energy $E_x = 7.12$ MeV in
$^{16}$O has been believed to enhance the $S$-factor of
$^{12}$C($\alpha$,$\gamma$)$^{16}$O. The enhancement seems to originate from
strong interference between 1$^-_1$ and 1$^-_2$ ($E_x\approx 9.6$ MeV) in the
vicinity of the $\alpha$-particle threshold. However, weak interference between
them and a resulting small $E$1 $S$-factor are exemplified with $R$-matrix
theory. Including a higher-order correction of the resonance parameters, the
present example appears to reproduce the experimental data consistently. It
would therefore be possible that the $E$1 $S$-factor is reduced at low
energies.
