By assuming a type of balance for length $\ell=87$ and non-trivial subgroups of multiplier groups of Legendre pairs (LPs) for length $\ell=85$, we find LPs of these lengths. We then study the power spectral density (PSD) values of m-compressions of LPs of length 5m. We also formulate a conjecture for Legendre pairs of lengths $\ell \equiv 0$ (mod 5) and demonstrate how it can be used to decrease the search space and storage requirements for finding such LPs. The newly found LPs decrease the number of integers in the range $\leq 200$ for which the existence question of LPs remains unsolved from 12 to 10.