In this work, we demonstrate a broadband tunable single-longitudinal-mode (SLM) random laser based on Rayleigh backscattering in a standard single-mode fiber. The wide tuning range of this SLM fiber laser over 1500-1570 nm is demonstrated with a linewidth of 4.5-30 kHz. The tuning is achieved using a tunable bandpass Fabry-Perot filter, and a semiconductor optical amplifier is used as the wide-bandwidth gain medium. The laser is able to operate in the S + C + L band.