Nuclear magnetic shielding tensor functions for H-2, HF, N-2, CO, and F-2 are calculated at the coupled-cluster singles and doubles level augmented by a perturbative correction for triple excitations [CCSD(T)]. The shielding constants for the lowest rovibrational states of these diatomics are obtained by solving the rovibrational Schrodinger equation with the finite-element method. For H, C, and F, absolute scales for the nuclear magnetic shielding constants have been obtained by combining computed diamagnetic shieldings with paramagnetic contributions deduced from measured spin-rotation constants and calculated rovibrational corrections. Since the experimental spin-rotation constants for N-2 and CO are inaccurate, shielding scales for N and O based on coupled cluster calculations are probably the most accurate available.