The first measurements of the polarized structure function σLT′ for the reaction 1H(e→,e′K+)Λ in the nucleon resonance region are reported. Measurements are included from threshold up to W=2.05 GeV for central values of Q2 of 0.65 and 1.00 GeV2, and nearly the entire kaon center-of-mass angular range. σLT′ is the imaginary part of the longitudinal-transverse response and is expected to be sensitive to interferences between competing intermediate s-channel resonances, as well as resonant and nonresonant processes. The results for σLT′ are comparable in magnitude to previously reported results from CLAS for σLT, the real part of the same response. An intriguing sign change in σLT′ is observed in the high Q2 data at W≈1.9 GeV. Comparisons to several existing model predictions are shown.