The polarized longitudinal-transverse structure function σLT′ measures the interference between real and imaginary amplitudes in pion electroproduction and can be used to probe the coupling between resonant and nonresonant processes. We report new measurements of σLT′ in the N(1440)12+ (Roper) resonance region at Q2=0.40 and 0.65GeV2 for both the π0p and π+n channels. The experiment was performed at Jefferson Lab with the CEBAF Large Acceptance Spectrometer (CLAS) using longitudinally polarized electrons at a beam energy of 1.515 GeV. Complete angular distributions were obtained and are compared to recent phenomenological models. The σLT′(π+n) channel shows a large sensitivity to the Roper-resonance multipoles M1− and S1− and provides new constraints on models of resonance formation.