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Let $\mathbb{I}$ denote an imaginary quadratic field or the field$\mathbb{Q}$ of rational numbers and $\mathbb{Z}_{\mathbb{I}}$ its ring ofintergers. We shall prove an explicit Baker type lower bound for$\mathbb{Z}_{\mathbb{I}}$-linear form of the numbers \begin{equation}\label{1}1,\ e^{\alpha_1},...,\ e^{\alpha_m},\quad m\ge 2, \end{equation} where$\alpha_0=0$, $\alpha_1,...,\alpha_m$, are $m+1$ different numbers from thefield $\mathbb{I}$. Our work gives gives some improvements to the existingexplicit versions of of Baker's work about exponential values at rationalpoints. In particilar, dependences on $m$ are improved.

An explicit Baker type lower bound of exponential values