The main goal of this work is to elucidate the effect of the inclusion of a rigid triaxial prolate primary body on the dynamics of the circular restricted three-body problem. The position, type, and linear stability of the coplanar points of equilibria are determined by numerical methods. Specifically, we conduct a systematic numerical study for elucidating the influence on the dynamics of the system of the triaxial prolate shape. Our findings are conclusive and indicate that the triaxiality parameters are very influential on the equilibria of the system, modifying not only the total number of equilibrium points but also their stability and nature. We also compare our findings with those of the previous paper of the series, where we have studied the case of triaxial oblate primary bodies.