In this paper, we characterize the Seifert matrices of $p$-periodic links whose quotients are 2-bridge links $C(2,n_1,-2,n_2,{cdots},n_r,(-2)^r)$ and give formulas for the signatures and determinants of the 3-periodic links of these kind in terms of $n_1$, $n_2$, ${cdots}$, $n_r$.