Summary: The generalized Mittag-Leffler function has been studied for arbitrary complex argument and parameters and . This function plays a fundamental role in the theory of fractional differential equations and numerous applications in physics. The Mittag-Leffler function interpolates smoothly between exponential and algebraic functional behaviour.
A numerical algorithm for its evaluation is developed. The algorithm is based on integral representations and exponential asymptotics. Results of extensive numerical calculations for in the complex -plane are reported. We find that all complex zeros emerge from the point for small . They diverge towards for and towards for . All the complex zeros collapse pairwise onto the negative real axis for . We introduce and study also the inverse generalized Mittag-Leffler function defined as the solution of the equation . We determine its principal branch numerically.