We provide an approximate zero S(g, L) for the hyperbolic Kepler's equation S − g arcsinh(S) − L = 0 for g ∈ (0, 1) and L ∈ [0, ∞). We prove, by using Smale's α-theory, that Newton's method starting at our approximate zero produces a sequence that converges to the actual solution S(g, L) at quadratic speed, i.e. if S n is the value obtained after n iterations, then |S n − S| ≤ 0.5 2 n −1 | S − S|. The approximate zero S(g, L) is a piecewisedefined function involving several linear expressions and one with cubic and square roots. In bounded regions of (0, 1) × [0, ∞) that exclude a small neighborhood of g = 1, L = 0, we also provide a method to construct simpler starters involving only constants.