On a refinement of the asymptotic formula for the Lebesgue constants

For the Lebesque constant of the classical Lagrange polynomial defined in the even number of nodes of interpolation, strict two-sided estimation is received. On this basis, an undefined value $O(1)$ is refined in the well-known asymptotic equality for the Lebesque constant. Two actual problems in the interpolation theory associated with the optimal choice of $O(1)$ are solved.