  Belinschi et al. [Adv. Math., 226 (2011), 3677--3698] proved that the normal
distribution is freely infinitely divisible. This paper establishes a certain
monotonicity, real analyticity and asymptotic behavior of the density of the
free L\'{e}vy measure. The monotonicity property strengthens the result in
Hasebe et al. [Int. Math. Res. Not. (2019), 1758--1787] that the normal
distribution is freely selfdecomposable.
