In this letter, we compose a new three-stage deep convolutional neural network (NTSDCN) for image demosaicking, and it consists of our proposed Laplacian energy-constrained local residual unit (LC-LRU) and a feature-guided prior fusion unit (FG-PFU). Specifically, the LC-LRU is used to refine the learning target of the specific residual blocks in the network and enhance the dominant information of the residual features. The FG-PFU is designed to guide the feature extraction of the red (R) and blue (B) channels by utilizing prior information from the reconstructed green (G) channel. In our proposed NTSDCN, we recover the G channel image in the first stage with the CFA image and reconstruct the R and B images in the second stage. Finally, we fine-tune the resulting R, G and B images in the third stage to compose a full-color RGB image. The experimental results show that our proposed method achieves better performance than the state-of-the-art methods. The code is available at https://github.com/wyannn/NTSDCN.