The Carmichael function $\lambda(n)$ is defined as the smallest positive integer $m$ such that $a^m = 1$ modulo $n$ for all integers $a$ coprime with $n$.
For example $\lambda(8) = 2$ and $\lambda(240) = 4$.

Define $L(n)$ as the smallest positive integer $m$ such that $\lambda(k) \ge n$ for all $k \ge m$.
For example, $L(6) = 241$ and $L(100) = 20\,174\,525\,281$.

Find $L(20\,000\,000)$. Give the last $9$ digits of your answer.