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The spectral‐energy (and luminosity) correlations in long gamma‐ray bursts 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  obs  peak   versus the fluence  F  or the peak flux  P . In a recent paper, we have started to investigate this problem considering all bursts with known redshift and spectral properties. Here, we consider instead all bursts with known   E  obs  peak  , irrespective of redshift, adding to those a sample of 100 faint  BATSE  bursts representative of a larger population of 1000 objects. This allows us to construct a complete, fluence‐limited, sample, tailored to study the selection/instrumental effects we consider. We found that the fainter  BATSE  bursts have smaller   E  obs  peak   than those of bright events. As a consequence, the   E  obs  peak   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 six per cent of these bursts are surely outliers of the   E  peak – E  iso   correlation (updated in this paper to include 83 bursts), since they are inconsistent with it for any redshift.   E  obs  peak   also correlates with the peak flux, 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  obs  peak – P   correlation defined by the  BATSE  bursts of our sample is significantly smaller than the   E  obs  peak – F   correlation of the same bursts, while for the bursts with known redshift the   E  peak – E  iso   correlation is tighter than the   E  peak – L  iso   one. Once a very large number of bursts with   E  obs  peak   and redshift will be available, we thus expect that the   E  peak – L  iso   correlation will be similar to that currently found, whereas it is very likely that the   E  peak – E  iso   correlation will become flatter and with a larger scatter.

Peak energy of the prompt emission of long gamma‐ray bursts versus their fluence and peak flux