There is no such thing as a "generic" lowpass filter: would it be all-pole, pole-zero, digital, if so IIR, FIR? To top it off, all these have their own topologies, so I'm afraid you're right with this part: "the question at hand is simply ill posed".
But, it looks like you're in the analog domain, so let's say there are two possibilities: all-pole and pole-zero. In these two cases, the generic transfer functions would look something like this:
$$H(s)=\frac{a_4 s^4+a_3 s^3 + a_2 s^2 + a_1 s + a_0}{b_4 s^4 + b_3 s^3 + b_2 s^2 + b_1 s + b_0}$$
For all-pole, the \$a_4\$ to \$a_1\$ terms would be zero, only \$a_0\$ remains. But the most generic way of representing any analog filter through its roots is:
$$\prod^N_{k=0}\frac{s-z_k}{s-p_k}$$
where z and p are the zeroes and poles, respectively, and they can be real or complex. As a side note, Butteroworth and Chebyshev (type I) are all-pole filters, with the particularity that Butterworth can be derived from a Chebyshev if the passband ripples are zero.
As it stands, I'm afraid your question cannot be answered. In general, making a filter is done by first stating the requirements: in frequency domain it's the cutoff frequencies, the attenuations, whether there are ripples or not, passband or stopband, etc or, if it's time-domain the linearity of the phase or the group delay. Then what particular topology to use, Sallen-Key, multiple-feedback, Friend, Delyannis, etc. So, there is a bit of work but it starts with the requirements.
