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In this paper we mainly focus on some determinants with Legendre symbolentries. Let $p$ be an odd prime and let $(\frac{\cdot}p)$ be the Legendresymbol. We show that $(\frac{-S(d,p)}p)=1$ for any $d\in\mathbb Z$ with $(\fracdp)=1$, and that $$\left(\frac{W_p}p\right)=\begin{cases}(-1)^{|\{0<k<\fracp4:\ (\frac kp)=-1\}|}&\text{if}\ p\equiv1\pmod4,\\(-1)^{\lfloor(p+1)/8\rfloor}&\text{if}\ p\equiv3\pmod4,\end{cases}$$ where$$S(d,p)=\det\left[\left(\frac{i^2+dj^2}p\right)\right]_{1\le i,j\le(p-1)/2}$$and $$W_p=\det\left[\left(\frac{i^2-((p-1)/2)!j}p\right)\right]_{0\lei,j\le(p-1)/2}.$$ We also pose some conjectures on determinants, one of whichstates that $(-1)^{\lfloor(p+1)/8\rfloor}W_p$ is a square when $p\equiv3\pmod4$.

On some determinants with Legendre symbol entries