On the roots of the equation $Z^{\mathrm{\prime }}(t)=0$

We have proved in this paper that the Lindel" of hypothesis generates essential contraction of distances between consecutive odd-order zeros of the function $Z^{\mathrm{\prime }}(t)$. This paper is the translation of the paper \cite{11} into the English except part 8 that we added in order to point out the I. M. Vinogradov’ scepticism on possibilities of the method of trigonometric sums.