We study the single longitudinal-spin asymmetry of dihadron production in semi-inclusive deep inelastic scattering process in which the transverse momentum of the final-state hadron pairs is integrated out. In Particular, we investigate origins of the $\sin \phi_R$ azimuthal asymmetry for which we take into account the coupling of the twist-3 distributions $h_L$ and the dihadron framgentation function (DiFF) $H_{1,ot}^{\sphericalangle}$ as well as the coupling of the helicity distribution $g_1$ and the twist-3 DiFF $\widetilde{G}^{\sphericalangle}$. To this end The unknown twist-3 dihadron fragmentation function $\widetilde{G}^{\sphericalangle}$ is calculated in a spectator model which is successful in describing the dihadron production in unpolarized process. We estimate the $\sin \phi_R$ asymmetry of dihadron production in SIDIS at the kinematics of COMPASS and compare it with the preliminary COMPASS data. Although the asymmetry is dominated by the $h_L H_{1}^{\sphericalangle}$ term, we find that the contribution from the $g_1\, \widetilde{G}^{\sphericalangle}$ term should also be taken into account in certain kinematical region.