This paper deals with the reconstruction of T1-T2 correlation spectra in Nuclear Magnetic Resonance (NMR) relaxometry. The ill-posed character and the large size of this inverse problem are the main difficulties to tackle. While maximum entropy is retained as an adequate regularization approach, the choice of an efficient optimization algorithm remains a challenging task. Our proposal is to apply a truncated Newton algorithm with three original features. Firstly, a theoretically well stated line search strategy suitable for the entropy function is applied to ensure the convergence of the algorithm. Secondly, an appropriate preconditioning structure based on a truncated singular value decomposition of the forward model matrix is used to speed up the algorithm convergence. Furthermore, we exploit the specific structures of the observation model and the Hessian of the criterion to reduce the computation cost of the algorithm. The performances of the proposed strategy are illustrated by means of synthetic and real data processing.