In this article, we study the zeta function $��_q$ associated to the Laplace operator $��_q$ acting on the space of the smooth $(0,q)$-forms with $q=0,\ldots,n$ on the complex projective space $\mathbb{P}^n(\mathbb{C})$ endowed with its Fubini-Study metric. In particular, we show that the values of $��_q$ at non-positive integers are rational. Moreover, we give a formula for $ \sum_{q\geq 0}(-1)^{q+1}q��_q'(0),$ the associated holomorphic analytic torsion.