Using a sample of 14 BeppoSAX and 74 Swift GRBs with measured redshift we tested the correlation between the intrinsic peak energy of the time-integrated spectrum, E_p,i, the isotropic-equivalent peak luminosity, L_p,iso, and the duration of the most
intense parts of the GRB computed as T_0.45 (Firmani correlation). For 41 out of 88 GRBs we could estimate all of the three required properties. Apart from 980425, which appears to be a definite outlier and notoriously peculiar in many respects, we used 40 GRBs to fit the correlation with the maximum likelihood method discussed by DAgostini, suitable to account for the extrinsic scatter in addition to the intrinsic uncertainties affecting every single GRB. We confirm the correlation. However, unlike the results by Firmani et al., we found that the correlation does have a logarithmic scatter comparable with that of the E_p,i-E_iso (Amati) correlation. We also find that the slope of the product L_p,iso T_0.45 is equal to ~0.5, which is consistent with the hypothesis that the E_p,i-L_p,iso-T_0.45 correlation is equivalent to the E_p,i-E_iso correlation (slope ~0.5). We conclude that, based on presently available data, there is no clear evidence that the E_p,i-L_p,iso-T_0.45 correlation is different (both in terms of slope and dispersion) from the E_p,i-E_iso correlation.
