A study of the mechanical characteristics and the low-temperature specific heat has been performed on a class of polymers whose crystallinity degree varies in a wide range, from a wholly amorphous to a highly crystalline structure. Typical features of the amorphous phase have been analyzed in order to shed further light on their correlations : the deviation from the exponentiality and the ''fragility'' obtained by modeling the mechanical alpha(a)-relaxation and the excess specific heat over the predictions of the Debye theory. It has been found that increasing crystallinity enhances the nonexponentiality of the alpha(a)-relaxation, and decreases both the fragility and the excess specific heat. This behavior is intrinsic to semicrystalline polymers. Consequently, the recently suggested correlation between the degree of fragility and the additional law-energy vibrations causing the low-temperature excess specific heat in glasses fails in semicrystalline polymers.