Three possibilities that come to mind but let's focus on this one:
simulate this circuit– Schematic created using CircuitLab
This schematic has a traditional pair of zeros and three poles. It also requires three terms, one each for lowpass, bandpass, and highpass. (The lowpass and highpass will be reflections/mirrors of each other.)
The denominator will have three roots. Optimally, two of them will be complex conjugates. The other will be real. But this isn't required. It's possible that all three are complex and there is no complex conjugate pair, at all. (Although it remains likely that two of the complex values are near to each other.)
The numerator will both real but not necessily located at the same place.
In general then, we'd prefer to find a complex conjugate pair in the denominator.
But this isn't always possible.
3rd order is a thing of its own.
So this is why I sugggest that there isn't really a standard form for 3rd order transfer functions.