<p>The work is devoted to finding an approximate analytical solution of the Local Induction Equation describing evolution of the vortex filament of a conical helical form. The solution for the radial coordinate of the filament is presented in implicit quadrature form. Analitical formulas for the two remaining coordinates, angular and axial ones are derived. The found approximate solution provides clear picture of the vortex filament dynamics in dependence on the problem parameters. This sollution allows one to omit the initial period during numerical integration of the Local Induction Equation.</p>