Let D be a square-free positive integer and let $K_D=mathbb{Q}(sqrt{-D})$ be the imaginary quadratic field. And let $h_D$ be the class number of the number field $K_D$. In this paper, we show the following: If D=l or 4l, where l is a prime number with $l{equiv}3$ (mod 4), then $h_D$ is odd.