For an integer $n\ge 3$ and a permutation $\sigma=(p_{1},p_{2},\cdots ,p_{n})$ of $\{1,2,\cdots , n\}$, we say $p_{l}$ is a $landmark$ point if $2\le l\le n-1$ and $(p_{l-1}-p_{l})(p_{l+1}-p_{l})>0$. For example , for $n=7$,\\
the permutation $(2,7,6,4,5,1,3)$ has four landmark points: $p_{2}=7$, $p_{4}=4$, $p_{5}=5$ and $p_{6}=1$. For a given $n\ge 3$ , let $L(n)$ denote the number of permutations of $\{1,2,\cdots ,n\}$ with exactly one landmark point. Find the maximum $n\ge 3$ for which $L(n)$ is a perfect square.