We present the X-ray luminosity function (XLF) for clusters of galaxies derived from the RASS1 Bright Sample. The sample, selected from the ROSAT All-Sky Survey in a region of 2.5 sr within the southern Galactic cap, contains 130 clusters with flux limits in the range ~ 3-4 x 10^-12 ergs/cm^2/s in the 0.5-2.0 keV band. A maximum-likelihood fit with a Schechter function of the XLF over the entire range of luminosities (0.045 - 28. x 10^44 ergs/s), gives alpha = 1.52 +/- 0.11, L_* = 3.80 +0.70 -0.55 x 10^44 ergs/s, and A = 5.07 +/- 0.45 x 10^-7 Mpc^-3 (10^44 ergs/s)^(alpha-1). We investigate possible evolutionary effects within the sample, out to our redshift limit (z ~ 0.3), finding no evidence for evolution. Our results are in good agreement with other local estimates of the XLF, implying that this statistic for the local universe is now well determined. Comparison with XLFs for distant clusters (0.3 < z < 0.6), shows that no evolution is present for L_X < 10^{44} ergs/s. However, we detect differences at the 3 sigma level, between our local XLF and the distant one estimated by Henry et al. for the EMSS sample. This difference is still present when considering the EMSS sample revised by Nichol et al.
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