Approximation of the longest common subsequence length for two long random strings

The expected value $E$ of the longest common subsequence of letters in two random words is considered as a function of the $\alpha=|A|$ of alphabet and of words lengths $m$ and $n$. It is assumed that each letter independently appears at any position with equal probability.
An approximate analitic expression for $E(\alpha, m, n)$ calculation is presented that allow to calculate the $E (m, n, \alpha) $ with an accuracy of $0.3$ percent for $ 64 \leqslant m + n \leqslant 65,536 $ and $ 1 <\alpha < 129$. The coefficients are selected by hand and can be refined. It is expected that the formula holds for each grater values of the argument with the same relative error.