Second order driving point impedance of RC network
What's the problem? I haven't checked the detail of your algebra but lets do a sanity check. At dc, the impedance of the network is infinite. So put s=0 into your equation and the denominator becomes zero, so you have infinite impedance.

At high frequencies, the impedance must be R3 plus the parallel combination of R1 and R2. So let the \$s^2\$ terms dominate in your equation and the result is k. Looking good.

On the back of an envelope I just got something which can be written as :-

$$Z=k.\frac{s^2+2\zeta\omega_ns+\omega_n^2}{s(s+a)}$$

... which looks just like your expession.

It might though be better to write it as :-

$$Z=k.\frac{(s+\omega_1)(s+\omega_2)}{s(s+\omega_3)}$$

since being a passive network, the roots must be real. You can of course plug the coefficients of your numerator into the quadratic formula to solve for \$\omega_1\$ and \$\omega_2\$ but I'll leave that to you!