Filter transfer function
The formula you propose is actually that of a brute-force approach. If I apply this technique to your network, I will transform the \$RC\$ front-end filter and the input voltage into a Thévenin generator whose output impedance is \$R_1||C_1\$. The brute-force transfer function is thus:



However, should you try to develop this expression, you will end up in an ugly arm-long formula with many cross-products, with the obvious risk of making a lot of errors (I surely would!). The best and easiest way is to apply the fast analytical techniques or FACTs. The principle is straightforward: determine the time constants of your circuit when the excitation source is set to zero. Here, it is a voltage source \$V_{in}\$ and setting it to 0 V means replacing it by a short circuit. When done, temporarily disconnect each of the energy-storing elements and "look" through its connections to determine the resistance driving it. The below drawing shows the approach:



You now inspect these little individual sketches - no algebra, just look at the drawing - and collect all the time constants while you alternately place one element in its dc state (a short or an open respectfully for an inductor or a cap) or in its hi-frequency state (the opposite). You then capture all these numbers in a Mathcad sheet or equivelent:



So you can see that you naturally obtain a denominator whose shape obeys that of a formalized form: \$D(s):1+b_1s+b_2s^2+b_3s^3\$. The trick is then to factor this polynomial recognizing a dominant low-frequency pole followed by a second-order polynomial. This is the right way to write the transfer function in a low-entropy way, having a resonant frequency immediately visible and a quality factor. Finally, if you compare the brute-force expression and the one obtained with the FACTs, they are identical:



You see the approach here. The FACTs are not only the simplest and often the fastest tool to get the symbolic answer, they naturally lead to a low-entropy (read factored) form where gains, poles and zeroes (if any) are immediately observable. Also, if at the end you observe a difference between the brute-force approach and the FACTs, you just need to fix a simple drawing to make things right. With the brute-force approach, you re-start from scratch and yell : )