Multi-sensor fusion-based localization technology has achieved high accuracy in autonomous systems. How to improve the robustness is the main challenge at present. The most commonly used LiDAR and camera are weather-sensitive, while the FMCW radar has strong adaptability but suffers from noise and ghost effects. In this paper, we propose a heterogeneous localization method of Radar on LiDAR Map (RoLM), which can eliminate the accumulated error of radar odometry in real-time to achieve higher localization accuracy without dependence on loop closures. We embed the two sensor modalities into a density map and calculate the spatial vector similarity with offset to seek the corresponding place index in the candidates and calculate the rotation and translation. We use the ICP to pursue perfect matching on the LiDAR submap based on the coarse alignment. Extensive experiments on Mulran Radar Dataset, Oxford Radar RobotCar Dataset, and our data verify the feasibility and effectiveness of our approach.