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 $L(100)=20174525281$.

Find $L(20000000)$. Give the last $9$ digits of your answer.