This paper applies the recent fast iterative neural network framework, Momentum-Net, using appropriate models to low-dose X-ray computed tomography (LDCT) image reconstruction. At each layer of the proposed Momentum-Net, the model-based image reconstruction module solves the majorized penalized weighted least-square problem, and the image refining module uses a four-layer convolutional neural network (CNN). Experimental results with the NIH AAPM-Mayo Clinic Low Dose CT Grand Challenge dataset show that the proposed Momentum-Net architecture significantly improves image reconstruction accuracy, compared to a state-of-the-art noniterative image denoising deep neural network (NN), WavResNet (in LDCT). We also investigated the spectral normalization technique that applies to image refining NN learning to satisfy the nonexpansive NN property; however, experimental results show that this does not improve the image reconstruction performance of Momentum-Net.