We study the problem of existence of weak right or left or strong coprime factorizations in H-infinity over the right half-plane of an analytic function defined in some subset of the right half-plane. We give necessary and sufficient conditions for the existence of such coprime factorizations in terms of an optimal control problem over the doubly infinite continuous time axis. In particular, we show that an equivalent condition for the existence of a strong coprime factorization is that both the control and the filter algebraic Riccati equation (of an arbitrary realization that need not be well-posed) have a solution (in general unbounded and not even densely defined) and that a coupling condition involving these two solutions is satisfied. The proofs that we give are partly based on corresponding discrete time results which we have recently obtained.