Upper bound for the product of nonhomogeneous linear forms

It is proved that for any unimodular lattice $\Lambda$ with homogeneous minimum $L>0$ and any set of real numbers $\alpha_1,\alpha_2,\dots,\alpha_n$ there exists a point ($y_1, y_2,\dots,y_n$) of $\Lambda$ such that
$$ \prod_{1\le i\le n}|y_i+\alpha_i|\le2^{-n/2_\gamma n}(1+3L^{8/(3^n)/(\gamma^{2/3}-2L^{8/(3^n)})})^{-n/2}, $$
where $\gamma^n=n^{n/(n-1)}$.