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A search for t-channel exchange of leptoquarks (LQs) is performed in dimuon and dielectron spectra using proton-proton collision data collected at TeV with the CMS detector at the CERN LHC. The data correspond to an integrated luminosity of 138 fb−1. Eight scenarios are considered, in which up or down quarks couple to muons or electrons via a scalar or vector LQ exchange, for dilepton invariant masses above 500 GeV. The LQ masses are probed up to 5 TeV, beyond a regime probed by previous pair-production and single-production searches. The differential distributions of dilepton events are fit to templates that model the nonresonant LQ exchange and various standard model background processes. Limits are set on LQ-fermion coupling strengths for scalar and vector LQ masses in the 1–5 TeV range at 95% confidence level, establishing stringent limits on first- and second-generation LQs.