The reaction pp ® K⁺ +(Lp) was measured at T_p = 1.953 GeV and Q = 0° with a high missing mass resolution in order to study the Lp final state interaction. The large final state enhancement near the Lp threshold can be described using the standard Jost-function approach. The singlet and triplet scattering lengths and effective ranges are deduced by fitting simultaneously the Lp invariant mass spectrum and the total cross section data of the free Lp scattering.