  Measuring cross-sections at the LHC requires the luminosity to be determined
accurately at each centre-of-mass energy $\sqrt{s}$. In this paper results are
reported from the luminosity calibrations carried out at the LHC interaction
point 8 with the LHCb detector for $\sqrt{s}$ = 2.76, 7 and 8 TeV
(proton-proton collisions) and for $\sqrt{s_{NN}}$ = 5 TeV (proton-lead
collisions). Both the "van der Meer scan" and "beam-gas imaging" luminosity
calibration methods were employed. It is observed that the beam density profile
cannot always be described by a function that is factorizable in the two
transverse coordinates. The introduction of a two-dimensional description of
the beams improves significantly the consistency of the results. For
proton-proton interactions at $\sqrt{s}$ = 8 TeV a relative precision of the
luminosity calibration of 1.47% is obtained using van der Meer scans and 1.43%
using beam-gas imaging, resulting in a combined precision of 1.12%. Applying
the calibration to the full data set determines the luminosity with a precision
of 1.16%. This represents the most precise luminosity measurement achieved so
far at a bunched-beam hadron collider.
