  A possibility of the realization of the p-wave spin-singlet superconductivity
($p$SS), whose gap function is odd both in momentum and in frequency, is
investigated by solving the gap equation with the phenomenological interaction
mediated by the antiferromagnetic spin fluctuation. The $p$SS is realized
prevailing over the d-wave singlet superconductivity ($d$SS) in the vicinity of
antiferromagnetic quantum critical pint (QCP) both on the paramagnetic and on
the antiferromagnetic sides. Off the QCP in the paramagnetic phase, however,
the $d$SS with line-nodes is realized as \textit{conventional} anisotropic
superconductivity. For the present $p$SS state, there is no gap in the
quasiparticle spectrum everywhere on the Fermi surface due to its odd
frequency. These features can give a qualitative understanding of the anomalous
behaviors of NQR relaxation rate on CeCu$_2$Si$_2$ or CeRhIn$_5$ where the
antiferromagnetism and superconductivity coexist on a microscopic level.
