For any two-sided jumping α-stable process, where 1 < α < 2, we find an explicit identity for the law of the first hitting time of the origin. This complements existing work in the symmetric case and the spectrally one-sided case; cf. Yano–Yano–Yor (2009) and Cordero (2010), and Peskir (2008) respectively. We appeal to the Lamperti–Kiu representation of Chaumont–Pantí–Rivero (2011) for real-valued self-similar Markov processes. Our main result follows by considering a vector-valued functional equation for the Mellin transform of the integrated exponential Markov additive process in the Lamperti–Kiu representation. We conclude our presentation with some applications.