  Let $l$ be a positive odd integer. Using Cilleruelo's method, we establish an
explicit lower bound $N_l$ depending on $l$ such that for all $n\geq N_l$,
$\prod_{k=1}^n (2k^2+l)$ is not a square. As an application, we determine all
values of $n$ such that $\prod_{k=1}^n (2k^2+l)$ is a square for certain values
of $l$.
