We analyse the population of near-Earth Long-Period Comets (LPCs) (perihelion distances q 10^3 yr). We have considered the sample of LPCs discovered during the period 1900-2009 and their estimated absolute total visual magnitudes H. For the period 1900-1970 we have relied upon historical estimates of absolute total magnitudes, while for the more recent period 1970-2009 we have made our own estimates of H based on Green's photometric data base and IAU Circulars. We have also used historical records for the sample of brightest comets (H < 4.5) covering the period: 1500-1899, based mainly on Vsekhsvyatskii, Hasegawa and Kronk catalogues. We find that the cumulative distribution of H can be represented by a three-modal law of the form log_{10}N_{<H} = C + alpha times H, where the C's are constants for the different legs, and alpha \simeq 0.28 +/- 0.10 for H < 4.0, alpha \simeq 0.56 +/- 0.10 for 4.0 <= H < 5.8, and alpha \simeq 0.20 +/- 0.02 for 5.8 <= H <8.6. The large increase of the slope of the second leg of the H-distribution might be at least partially attributed to splitting of comet nuclei leading to the creation of two or more daughter comets. The cumulative H-distribution tends to flatten for comets fainter than H <= 8.6. LPCs fainter than H <= 12 (or diametres D \lesssim 0.5 km) are extremely rare, despite several sky surveys of near-Earth objects implemented during the last couple of decades, suggesting a minimum size for a LPC to remain active. We also find that about 30 % of all LPCs with q < 1.3 AU are new (original bound energies 0 < E_{or} < 10^{-4} AU^{-1}), and that among the new comets about half come from the outer Oort cloud (energies 0 \lesssim E_{or} \lesssim 0.3 times 10^{-4} AU^{-1}), and the other half from the inner Oort cloud (energies 0.3 times 10^{-4} \lesssim E_{or} \lesssim 10^{-4}AU^{-1}).