The paper is devoted to the proof of the following theorem: Let \(v_ j\) \((j=1,2,\ldots,k)\) be a zero of the \(k\)-th partial sum of \(e^ x\): \(P_ k(z)+1+x+{x^ 2\over 2!}+\cdots+{x^ k\over k!}\) and \(P_ k(v_ j)=0\). We have \[ \sum^ k_{j=1}{e^{-2v_ j}\over v_ j^ 2}=\begin{cases} 2k+4+{2\over k}+O(k^{-3/2}), & \text{ if } k \text{ is odd} \\ 2k+4-{6\over k}+O(k^{-3/2}), & \text{ if } k \text{ is even}. \end{cases} \] The studied sums are connected with certain mean-values of the Riemann zeta-function. The obtained theorem is in some sense an improvement of previous results.