Electron-ion recombination of Be-like 40Ar14+ has been measured by employing the electron-ion merged-beams method at the cooler storage ring CSRm. The measured absolute recombination rate coefficients for collision energies from 0 to 60 eV are presented, covering all dielectronic recombination (DR) resonances associated with 2s 2 → 2s2p core transitions. In addition, strong trielectronic recombination (TR) resonances associated with 2s 2 → 2p 2 core transitions were observed. Both DR and TR processes lead to series of peaks in the measured recombination spectrum, which have been identified by the Rydberg formula. Theoretical calculations of recombination rate coefficients were performed using the state-of-the-art multi-configuration Breit-Pauli atomic structure code AUTOSTRUCTURE to compare with the experimental results. The plasma rate coefficients for DR+TR of Ar14+ were deduced from the measured electron-ion recombination rate coefficients in the temperature range from 103 to 107 K, and compared with calculated data from the literature. The experimentally derived plasma rate coefficients are 60% larger and 30% lower than the previously recommended atomic data for the temperature ranges of photoionized plasmas and collisionally ionized plasmas, respectively. However, good agreement was found between experimental results and the calculations by Gu and Colgan et al. The plasma rate coefficients deduced from experiment and calculated by the current AUTOSTRUCTURE code show agreement that is better than 30% from 104 to 107 K. The present results constitute a set of benchmark data for use in astrophysical modeling.