This paper describes and validates an advanced synthesis design process of Frequency Selective Surfaces (FSS) with elliptic band-stop response. A systematic procedure based on the Generalized Chebyshev function and the extracted-pole technique enables to control the position of the transmission zeros and the attenuation level, to obtain and equiripple rejection response. A systematic process is followed to obtain the lumped LC values of the resonators circuits extracted as poles and the impedance inverters. Then, equivalent dipoles and transmission lines are obtained to carry out the electromagnetic design at normal incidence for a linearly polarized field. The impact of the high order modes of the periodic structure on the electrical response of the FSS, which can be relevant due to the stringent selected specifications, have been also analyzed. A four order band-stop filter with 3 (GHz) bandwidth centered at 30 (GHz) and attenuation 50 (dB) has been designed considering three different implementations: two filters using vacuum as transmission line with different connection lengths and a third one using dielectric substrate to enable its manufacturing. In order to verify the design procedure by experimental results, the third filter with printed dipoles in the dielectric substrate was manufactured and measured, validating the developed process.