In this paper, we study the linear stability of an infinite vortex ring array with respect to the pairing instability, using a spectral code. The base flow solution, obtained after a short relaxation process, is composed of rings with a Gaussian azimuthal vorticity profile. The temporal stability properties are first obtained and compared to the theoretical predictions obtained by Levy and Forsedyke (1927 Proc. R. Soc. Lond. A 114 594-604). The spatio-temporal evolution of a localized perturbation is then computed. The growth rate σ ≤ft( v \right) of the perturbation in the frame moving at the speed v is obtained for all v. The variation of σ ≤ft( v \right) with respect to the parameters of the flow is provided.