We estimate a gamma-ray burst (GRB) formation rate based on the new relation between the spectral peak energy (E-p) and the peak luminosity. The new relation is derived by combining the data of E-p and the peak luminosities by BeppoSAX and BATSE, and it looks considerably tighter and more reliable than the relations suggested by the previous works. Using the new E-p-luminosity relation, we estimate redshifts of the 689 GRBs without known distances in the BATSE catalog and derive a GRB formation rate as a function of the redshift. For the redshift range of 0 less than or equal to z less than or equal to 2, the GRB formation rate increases and is well correlated with the star formation rate, while it keeps constant toward z similar to 12. We also discuss the luminosity function and the redshift dependence of the intrinsic luminosity ( luminosity evolution).