  In this paper we investigate determinants whose entries are linear
combinations of Legendre symbols. We deduce some new results in this direction;
for example, we prove that for any prime $p\equiv3\pmod4$ we have
$$\det\left[x+\left(\frac{j-k}p\right)+\left(\frac jp\right)-\left(\frac
kp\right)\right]_{0\le j,k\le(p-1)/2}=4,$$ where $(\frac{\cdot}p)$ is the
Legendre symbol. We also pose many conjectures for further research. For
example, for any prime $p>3$ we conjecture that \begin{align*}&\
\det\left[\left(\frac{j+k}p\right)+\left(\frac{j-k}p\right)+\left(\frac{jk}p\right)\right]_{1\le
j,k\le(p-1)/2} \\=&\ \begin{cases}(\frac 2p)p^{(p-5)/4}&\text{if}\
p\equiv1\pmod4, \\(-1)^{(h(-p)-1)/2}(1-(2-(\frac
2p))h(-p))p^{(p-3)/4}&\text{if}\ p\equiv3\pmod4, \end{cases}\end{align*} where
$h(-p)$ is the class number of the imaginary quadratic field $\mathbb
Q(\sqrt{-p})$.
