Have I obtained the correct transfer functions for the following RCL circuit?
I won't show what you have wrong, since you're saying that you're practising, but it doesn't make much sense to not use LTspice to verify yourself as opposed to just using it as a screenshot provider. As I said in the comments, this is how you could do it (BTW, there was a typo I missed in the Laplace expression):

Note that Laplace expressions can use ^ as exponentiation, but only here, otherwise the double asterisk ** is used. Two minor &quot;gotcha&quot;s:

The source is a voltage source so as long as it doesn't have any resistance in the way (be it external, like R1, or internal, like Rser), the input voltage will be unaffected therefore it can be reused for the E1.
Inductors and capacitors have some default parasitics specified, L has 1 mΩ series resistance, and C a 1/Gmin parallel resistance, so depending on the amount of bandwidth you're simulating, you may see some effects. To be sure there are none, RClick on the L or C and set all the parasitics to zero.


I find it difficult to follow your math, so here's how it should go, using your approach (there are many ways to do it):
$$\begin{align}
Z_c&amp;=\dfrac{1}{sC}+R_2 \\
&amp;=\dfrac{sR_2C+1}{sC} \tag{1} \\
Z_o&amp;=\dfrac{1}{\dfrac{1}{sL}+\dfrac{sC}{sR_2C+1}} \\
&amp;=\dfrac{s^2R_2LC+sL}{s^2LC+sR_2C+1} \tag{2} \\
H(s)&amp;=\dfrac{Z_o}{Z_o+R_1}=\dfrac{\mathrm{Num}\Bigl(H(s)\Bigr)}{\mathrm{Den}\Bigl(H(s)\Bigr)} \tag{3} \\
\mathrm{Num}\Bigl(H(s)\Bigr)&amp;=\dfrac{s^2R_2LC+sL}{s^2LC+sR_2C+1} \tag{4} \\
\mathrm{Den}\Bigl(H(s)\Bigr)&amp;=\dfrac{s^2R_2LC+sL}{s^2LC+sR_2C+1}+R_1 \\
&amp;=\dfrac{s^2LC(R_1+R_2)+s(R_1R_2C+L)+R_1}{s^2LC+sR_2C+1} \tag{5} \\
\Rightarrow H(s)&amp;=\dfrac{s^2R_2LC+sL}{s^2LC(R_1+R_2)+s(R_1R_2C+L)+R_1} \tag{6}
\end{align}$$
I have skipped basic math steps, I'm sure you can follow the thread. And this is the test:

The red trace has a slightly different phase than the Laplace expression and that's because of the 2nd point mentioned above: the default 1 mΩ series resistor for the inductance (left on purpose).