The generalized Mittag-Leffler function is defined by the sum
when this reduces to the ordinary Mittag-Leffler fuction . The authors point out that the solution of the eigenvalue equation , where denotes the Riemann-Liouville fractional derivative, is solved by
The aim of this paper is to study numerically the function in the complex plane. All calculations are presented for the particular case , . The authors employ the contour integral representation
where is a path that lies outside the disc . The portion of the plane studied is , which is divided into a grid consisting of points. Three-dimensional plots of the real and imaginary parts of are presented. A contour plot of the real and imaginary parts is also given, which includes the first pair of complex conjugate zeros of situated at approximately .