In this paper we study some conjectures on determinants with Jacobi symbol entries posed by Z.-W. Sun. For any positive integer n≡3(mod4), we show that (6,1)_n=[6,1]_n=(3,2)_n=[3,2]_n=0 and (4,2)_n=(8,8)_n=(3,3)_n=(21,112)_n=0 as conjectured by Sun, where [Formula presented] and [Formula presented] with [Formula presented] the Jacobi symbol. We also prove that (10,9)_p=0 for any prime p≡5(mod12), and [5,5]_p=0 for any prime p≡13,17(mod20), which were also conjectured by Sun. Our proofs involve character sums over finite fields.