Can you rewrite this function to be independent of the variable $p$?

Question

A while ago Roger L. Bagula came up with the idea to study this function:

(*Mathematica 8 start*)
f[p_, n_] = 
 Zeta[1/2 + I*(1 - p)*Im[ZetaZero[n - 1]] + I*p*Im[ZetaZero[n]]]
b = Table[
  ParametricPlot[{Re[f[p, n]], Im[f[p, n]]}, {p, 0, 1}], {n, 127, 128}]
(*Show[b,PlotRange->All]*)
(*end*)

Or in latex:

$$f(\text{p},\text{n})=\zeta \left(i (1-p) \Im\left(\rho _{n-1}\right)+i p \Im\left(\rho _n\right)+\frac{1}{2}\right)$$

Is it possible to rewrite $f(\text{p},\text{n})$ as a function $f(\text{n})$ which is independent of the variable $p$?

Please don't confuse the similar looking $p$ and $\rho$ in the formula.