Estimats of the inhomogeneous arithmetical minimum of the product of linear forms

Further refinements of Chebotarev type estimates are obtained for the inhomogeneous arithmetic minimum $M_n$ of a lattice $\Lambda$ of determinant $d(\Lambda)$ in the inhomogeneous Minkowski conjecture. In particular, it is proved that for every $n_0\geq2$ there exists an effectively computed constant $c=c(n_0)$ for which
$$ M_n\leq2^{-n/2}(cn^{-1/2}\log^{1/2}n)d(\Lambda). $$