Kostant partition function for $\mathfrak{sp}_4(\mathbb C)$

In this note, we obtain exact values of the partition function of Kostant for the simple Lie algebra $\mathfrak{sp}_4(\mathbb C)$. Using the values of the partition function, we can find the weight multiplicities of irreducible representations of $\mathfrak{sp}_4(\mathbb C)$ by a simple computation.