Axial inductor between the plates of a cylindrical capacitor
If the inductor is of negligible size and is connected to the plates (one side of the inductor to one plate, the other to the other plate), it forms a "tank" circuit, resonant at frequency \$f_0\$= \$1\over 2\pi\sqrt{LC}\$. 

It doesn't really matter topologically whether the connection to the plates is at any particular place, so for ideal components, there's no difference. 

In reality, the inductor would have some effect on the capacitance and a real inductor will have some distributed capacitance internally (between turns). 

The physical arrangement imagined here is a metal tube with a coaxial rod inside (cylindrical capacitor) with the inductor connected between the inside of the tube and the external surface of the rod. 







If it was not connected, it would depend on the way it was constructed, but mostly likely for most frequencies it would just increase the capacitance a bit because it is reducing the effective separation of the plates by virtue of inserting a conductor.