  We combine perturbation theory with analytic and numerical bootstrap
techniques to study the critical point of the long-range Ising (LRI) model in
two and three dimensions. This model interpolates between short-range Ising
(SRI) and mean-field behaviour. We use the Lorentzian inversion formula to
compute infinitely many three-loop corrections in the two-dimensional LRI near
the mean-field end. We further exploit the exact OPE relations that follow from
bulk locality of the LRI to compute infinitely many two-loop corrections near
the mean-field end, as well as some one-loop corrections near SRI. By including
such exact OPE relations in the crossing equations for LRI we set up a very
constrained bootstrap problem, which we solve numerically using SDPB. We find a
family of sharp kinks for two- and three-dimensional theories which compare
favourably to perturbative predictions, as well as some Monte Carlo simulations
for the two-dimensional LRI.
