We study the time-dependent and the stationary properties of the linear Glauber model in a d-dimensional hypercubic lattice. This model is equivalent to the voter model with noise. By using the Green function method, we get exact results for the two-point correlations from which the critical behavior is obtained. For vanishing noise the model becomes critical with exponents beta=0, gamma=1, and nu=1/2 for d > or =2, with logarithmic corrections at the upper critical dimension d(c)=2, and beta=0, gamma=1/2, and nu=1/2 for d=1. We show that the model can be mapped into a particular reaction-diffusion model.