We have obtained low-resolution optical (0.7–0.98 μm) and near-infrared (1.11–1.34 μm and 0.8–2.5 μm) spectra of 12 isolated planetary-mass candidates (J = 18.2–19.9 mag) of the 3 Myr σ Orionis star cluster with the aim of determining the spectroscopic properties of very young, substellar dwarfs and assembling a complete cluster mass function. We have classified our targets by visual comparison with high- and low-gravity standards and by measuring newly defined spectroscopic indices. We derived L0–L4.5 and M9–L2.5 using high- and low-gravity standards, respectively. Our targets reveal clear signposts of youth, thus corroborating their cluster membership and planetary masses (6–13 M Jup). These observations complete the σ Orionis mass function by spectroscopically confirming the planetary-mass domain to a confidence level of ∼75%. The comparison of our spectra with BT-Settl solar metallicity model atmospheres yields a temperature scale of 2350–1800 K and a low surface gravity of log g ≈ 4.0 [cm s‑2], as would be expected for young planetary-mass objects. We discuss the properties of the cluster’s least-massive population as a function of spectral type. We have also obtained the first optical spectrum of S Ori 70, a T dwarf in the direction of σ Orionis. Our data provide reference optical and near-infrared spectra of very young L dwarfs and a mass function that may be used as templates for future studies of low-mass substellar objects and exoplanets. The extrapolation of the σ Orionis mass function to the solar neighborhood may indicate that isolated planetary-mass objects with temperatures of ∼200–300 K and masses in the interval 6–13 M Jup may be as numerous as very low-mass stars.