  We investigate, using numerical methods, the convergence dynamics of the
Sitnikov problem with non-spherical primaries, by applying the Newton-Raphson
(NR) iterative scheme. In particular, we examine how the oblateness parameter
$A$ influences several aspects of the method, such as its speed and efficiency.
Color-coded diagrams are used for revealing the convergence basins on the plane
of complex numbers. Moreover, we compute the degree of fractality of the
convergence basins on the complex space, as a relation of the oblateness, by
using different computational tools, such the fractal dimension as well as the
(boundary) basin entropy.
