Against the background of 6G communication and big data, more and more attention have been paid to the communication reliability and resource efficiency. Reconfigurable intelligent surface (RIS) is envisioned as a potential technology to improve the communication environment by controlling the electromagnetic wave propagation. A majority of related researches use the central limit theorem (CLT) to implement the analytical evaluation, which results in inaccuracy for the case with small number of reflective elements. In this letter, we investigate the outage behavior of RIS-enabled downlink cellular networks where exist several device-to-device (D2D) pairs under the general fading channels, i.e., Nakagami-m fading channels. We propose a comprehensive solution to evaluate the outage probability for all the cases with different numbers of reflective elements. Our solution utilizes the multivariate Fox's H-function and proposes the outage expressions in closed form. Finally, The accuracy of the closed-form outage probability is verified in simulation section for different cases of system configurations.