Context. The fundamental properties of brown dwarfs evolve with age. Models describing the evolution of luminosities and effective temperatures, among other physical parameters, can be empirically constrained using brown dwarfs of various masses in star clusters of well-determined age and metallicity. Aims: We aim to carry out a spectroscopic and photometric characterization of low-mass brown dwarfs in the ~120 Myr old Pleiades open cluster. Methods: We obtained low-resolution, near-infrared spectra of the J = 17.4-18.8 mag candidate L-type brown dwarfs PLIZ 28 and 35, BRB 17, 21, 23, and 29, which are Pleiades members by photometry and proper motion. We also obtained spectra of the well-known J = 15.4-16.1 mag late M-type cluster members PPl 1, Teide 1, and Calar 3. Results: We find that the first six objects have early- to mid-L spectral types and confirm previously reported M-types for the three other objects. The spectra of the L0-type BRB 17 and PLIZ 28 present a triangular H-band continuum shape, indicating that this peculiar spectral feature persists until at least the age of the Pleiades. We add to our sample 36 reported M5-L0-type cluster members and collect their IC- and UKIDSS ZYJHK-band photometry. We confirm a possible interleaving of the Pleiades and field L-type sequences in the JHK absolute magnitude versus spectral type diagrams, and quantify marginally redder Pleiades J-K colours, by 0.11 ± 0.20 mag, possibly related to both reddening and youth. Using field dwarf bolometric correction - and effective temperature - spectral type relations, we obtain the Hertzsprung-Russell diagram of the Pleiades sample. Theoretical models reproduce the spectral sequence at M5.5-9, but appear to overestimate the luminosity or underestimate the effective temperature at L0-5. Conclusions: We classify six faint Pleiades brown dwarfs as early to mid L-type objects using low-resolution near-infrared spectra. We compare their properties to field dwarfs and theoretical models and estimate their masses to be in the range 0.025-0.035 M&sun;.