We present a linearly implicit and structure-preserving scheme to solve the space-fractional Ginzburg–Landau–Schrödinger equation. The fully discrete scheme is obtained by combining the modified leap-frog method in the temporal direction and the finite difference methods in the spatial direction. It is shown that the scheme can be unconditionally energy-stable. In particular, the equation becomes the space-fractional Schrödinger equation. Then, the scheme can keep both the discrete mass and energy conserved. Moreover, convergence of the scheme is obtained. Numerical experiments are performed to confirm the theoretical results.