  LS 5039 is a gamma-ray binary system observed in a broad energy range, from
radio to TeV energies. The binary system exhibits both flux and spectral
modulation as a function of its orbital period. The X-ray and very-high-energy
(VHE, E > 100 GeV) gamma-ray fluxes display a maximum/minimum at
inferior/superior conjunction, with spectra becoming respectively
harder/softer, a behaviour that is completely reversed in the high-energy
domain (HE, 0.1 < E < 100 GeV). The HE spectrum cuts off at a few GeV, with a
new hard component emerging at E > 10 GeV that is compatible with the
low-energy tail of the TeV emission. The low 10 - 100 GeV flux, however, makes
the HE and VHE components difficult to reconcile with a scenario including
emission from only a single particle population. We report on new observations
of LS 5039 conducted with the High Energy Stereoscopic System (H.E.S.S.)
telescopes from 2006 to 2015. This new data set enables for an
unprecedentedly-deep phase-folded coverage of the source at TeV energies, as
well as an extension of the VHE spectral range down to ~120 GeV, which makes LS
5039 the first gamma-ray binary system in which a spectral overlap between
satellite and ground-based gamma-ray observatories is obtained.
