We analyse the population of near-Earth long-period comets (LPCs; perihelion distances q < 1.3 au and orbital periods P > 103 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 the Vsekhsvyatskii, Hasegawa and Kronk catalogues. We find that the cumulative distribution of H can be represented by a three-modal law of the form log10N< H=C+αH, where the C values are constants for the different legs, and α≃ 0.28 ± 0.10 for H < 4.0, α≃ 0.56 ± 0.10 for 4.0 ≤H < 5.8, and α≃ 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 diameters D≲ 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 an LPC to remain active. We also find that about 30 per cent of all LPCs with q < 1.3 au are new (original bound energies 0 < Eor < 10−4 au−1), and that among the new comets about half come from the outer Oort cloud (energies 0 ≲Eor≲ 0.3 × 10−4 au−1), and the other half from the inner Oort cloud (energies 0.3 × 10−4≲Eor≲ 10−4 au−1).