Help deriving the transfer function of an LC circuit?
If you do the algebra (it's amazing what folk will do in lock-down of course) you get the following transfer function: -

$$H(s) = \dfrac{1}{s^3LC_1C_2R + s^2LC_2 + sR(C_1+C_2) + 1}$$

And just to demonstrate that is correct I used micro-cap 12 and the following circuit: -



Inside the red box is the input AC voltage that can be swept from low to high frequencies. To the left of that is the conventional circuit formed by R, L, C1 and C2.

To the right we have micro-caps 12's Laplace function solver. Both circuits are fed from V1 (red box).

Comparing both bode plots we get identical overlapping results: -



If this helps then that's good. If you require the derivation of the above TF then ask. I don't suppose it hurts to post my reasonably legible hand calcs: -