  We analyse the population of near-Earth Long-Period Comets (LPCs) (perihelion
distances q < 1.3 AU and orbital periods P > 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}).
