Estimates from below are obtained for polynomials with integral coefficients in the values of certain Siegel $G$ -functions at the algebraic points of a special form. In particular, it is proved that if $\alpha_1,\dots,\alpha_s$ ( $\alpha_1\cdots\alpha_s\ne0$ ) are pairwise distinct algebraic numbers, $q$ is a natural number, and $P(x_1,\dots,x_s)\not\equiv0$ is a polynomial with integral coefficients of degree not greater than $d$ and height not exceeding $H$ , then for $q>q_0(d,\alpha_1,\dots,\alpha_s)$ we have $$\Bigl|P\Bigl(\ln\Bigl(1+\frac{\alpha_1}q\Bigr),\dots,\ln\Bigl(1+\frac{\alpha_s}q\Bigr)\Bigr)\Bigr|>q^{-\lambda}H^{-\mu}, $$where the constants$q_0$and$\mu$can be effectively computed.Bibliography: 17 titles.