We use the renormalization group method to study the stochastic Navier-Stokes equation with a random force correlator of the form k ^4-d-2ε in a d-dimensional space in connection with the problem of constructing a 1/d-expansion and going beyond the framework of the standard ε-expansion in the theory of fully developed hydrodynamic turbulence. We find a sharp decrease in the number of diagrams of the perturbation theory for the Green's function in the large-d limit and develop a technique for calculating the diagrams analytically. We calculate the basic ingredients of the renormalization group approach (renormalization constant, β-function, fixed-point coordinates, and ultraviolet correction index ω) up to the order ε ³ (three-loop approximation). We use the obtained results to propose hypothetical exact expressions (i.e., not in the form of ε-expansions) for the fixed-point coordinate and the index ω.