  A correlation between the peak luminosity and the peak energy has been found
by Yonetoku et al. as $L_{p}\propto E_{p,i}^{2.0}$ for 11 pre-Swift long
gamma-ray bursts. In this study, for a greatly expanded sample of 148 long
gamma-ray bursts in the Swift era, we find that the correlation still exists,
but most likely with a slightly different power-law index, i.e., $L_{p}\propto
E_{p,i} ^{1.7}$. In addition, we have collected 17 short gamma-ray bursts with
necessary data. It is found that the correlation of $L_{p}\propto E_{p,i}
^{1.7}$ also exists for this sample of short events. It is argued that the
radiation mechanism of both long and short gamma-ray bursts should be similar,
i.e., of quasi-thermal origin caused by the photosphere and the dissipation
occurring very near the central engine. Some key parameters of the process are
constrained. Our results suggest that the radiation process of both long and
short bursts may be dominated by thermal emission, rather than the single
synchrotron radiation. This might put strong physical constraints on the
theoretical models.
