Measuring cross-sections at the LHC requires the luminosity to be determined accurately at each centre-of-mass energy s. In this paper results are reported from the luminosity calibrations carried out at the LHC interaction point 8 with the LHCb detector for s = 2.76, 7 and 8 TeV (proton-proton collisions) and for sNN = 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 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.
