The spectral-energy and (luminosity) correlations in long GRBs are being hotly debated to establish, first of all, their reality against possible selection effects. These are best studied in the observer planes, namely the peak energy E_peak_obs vs the fluence F or the peak flux P. In a recent paper we started to attack this problem considering all GRBs with known z and spectral properties. Here we consider instead all bursts with known E_peak_obs, irrespective of z, adding to those a sample of 100 faint BATSE bursts representative of a larger population. This allows us to construct a complete, fluence limited, sample, to study the selection/instrumental effects. We found that fainter bursts have smaller E_peak_obs than those of bright events. As a consequence, the E_peak_obs of these bursts is correlated with the fluence, though with a slope flatter than that defined by bursts with z. Selection effects, which are present, are shown not to be responsible for the existence of such a correlation. About 6% of these bursts are surely outliers of the E_peak-E_iso correlation (updated to include 83 bursts), since they are inconsistent with it for any z. E_peak_obs correlates also with P, with a slope similar to the E_peak-L_iso correlation.In this case there is only one sure outlier.The scatter of the E_peak_obs-P correlation defined by the BATSE bursts of our sample is smaller than the E_peak_obs-F correlation of the same bursts, while for the bursts with known z the E_peak-E_iso correlation is tighter than the E_peak-L_iso one. Once a very large number of bursts with E_peak_obs and z will be available, we thus expect that the E_peak-L_iso correlation will be similar to that currently found, whereas it is likely that the E_peak-E_iso correlation will become flatter and with a larger scatter.
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