Consider a honey bee's honeycomb where each cell is a perfect regular hexagon with side length $1$.

One particular cell is occupied by the queen bee.
For a positive real number $L$, let $\text{B}(L)$ count the cells with distance $L$ from the queen bee cell (all distances are measured from centre to centre); you may assume that the honeycomb is large enough to accommodate for any distance we wish to consider.
For example, $\text{B}(\sqrt{3})=6$, $\text{B}(\sqrt{21})=12$ and $\text{B}(111111111)=54$.

Find the number of $L\le 5\times {10}^{11}$ such that $\text{B}(L)=450$.