6.44 For a prime $p \geqslant 3$, define
$$
F(p)=\sum_{k=1}^{\frac{p-1}{2}} k^{120}, \quad f(p)=\frac{1}{2}-\left\{\frac{F(p)}{p}\right\},
$$

where $\{x\}=x-[x]$ denotes the fractional part of $x$. Find the value of $f(p)$.
(China National Team Selection Test, 1993)