The double differential cross sections of the Drell-Yan lepton pair (l(+)l(-), dielectron or dimuon) production are measured as functions of the invariant mass m(ll), transverse momentum p(T)(ll), and phi(eta)*. The phi(eta)* observable, derived from angular measurements of the leptons and highly correlated with p(T)(ll), is used to probe the low-p(T)(ll) region in a complementary way. Dilepton masses up to 1 TeV are investigated. Additionally, a measurement is performed requiring at least one jet in the final state. To benefit from partial cancellation of the systematic uncertainty, the ratios of the differential cross sections for various m(ll) ranges to those in the Zmass peak interval are presented. The collected data correspond to an integrated luminosity of 36.3 fb(-1) of proton-proton collisions recorded with the CMS detector at theLHCat a centre-of-mass energy of 13 TeV. Measurements are compared with predictions based on perturbative quantum chromodynamics, including soft-gluon resummation.