  The planar linear restricted four-body problem is used in order to determine
the Newton-Raphson basins of convergence associated with the equilibrium
points. The parametric variation of the position as well as of the stability of
the libration points is monitored when the values of the mass parameter $b$ as
well as of the angular velocity $\omega$ vary in predefined intervals. The
regions on the configuration $(x,y)$ plane occupied by the basins of attraction
are revealed using the multivariate version of the Newton-Raphson iterative
scheme. The correlations between the attracting domains of the equilibrium
points and the corresponding number of iterations needed for obtaining the
desired accuracy are also illustrated. We perform a thorough and systematic
numerical investigation by demonstrating how the parameters $b$ and $\omega$
influence the shape, the geometry and of course the fractality of the
converging regions. Our numerical outcomes strongly indicate that these two
parameters are indeed two of the most influential factors in this dynamical
system.
