The specific shear viscosity, eta /s, of the quark-gluon plasma formed in ultrarelativistic heavy-ion collisions at RHIC and LHC is estimated based on the progressive longitudinal broadening of transverse momentum two-particle correlators, G_2, reported as a function of collision centrality by the STAR and ALICE experiments. Estimates are computed as a function of collision centrality using the Gavin ansatz which relates the G_2 longitudinal broadening to the specific shear viscosity. Freeze out times required for the use of the ansatz are computed using a linear fit of freeze out times reported as a function of the cubic root of the charged particle pseudorapidity density ({mathrm{d}}N_{mathrm{ch}}/deta )^{1/3}. Estimates of eta /s based on ALICE data exhibit little to no dependence on collision centrality at LHC energy, while estimates obtained from STAR data hint that eta /s might be a function of collision centrality at top RHIC energy.