$\newcommand{\oh}{\mathcal{O}} \newcommand{\ch}{\mathrm{ch}} \newcommand{\td}{\mathrm{td}}$Let $X$ be a Fano threefold of index $1$ and degree $d$ with ample class $H$, so $K_X=-H$ and $H^3=d$, and let $ i : L \hookrightarrow X$ be a line (with Hilbert polynomial $1+t$) in $X$. I have read that $\ch(i_* \oh_L) = L + \frac{1}{2}P$ where $L$ and $P$ are the classes of a line and a point in the cohomology ring $H^*(X, \mathbb{Q})$, respectively (note that $dL=H^2$). I'm trying to show this, but I'm making a mistake somewhere in my reasoning - my question is where?
Note that $i$ is proper, so $i_!=i_* $. Then by Grothendieck-Riemann-Roch we have $$ \ch(i_* \oh_L) \td(X) = i_*(\ch(\oh_L) \td (L)) . $$ I know that the Todd class of such $X$ can be given by $\td(X) = (1, \frac{1}{2}H, \frac{1}{12}(H^2 + c_2(X)), \frac{1}{24} H \cdot c_2(X))$. I also know that we have $H \cdot c_2(X) = 24$ (but I think I should put some class after $24$ here - perhaps $24P$?), so the last entry of the Todd class of $X$ is $1$ (or $P$?). Also, $\ch(\oh_L) = 1$ and since $L$ is an algebraic curve with $L \cong \mathbb{P}^1$, we have $\td(L) = 1 + c_1(L) = 1 - K_L = 1 + 2P$ (where here, $P$ is a point in $L$ as opposed to in $X$).
We also know that $i_* \oh_L$ is a sheaf on $X$ supported in codimension $2$, so it follows that $\mathrm{rk}(i_* \oh_L ) = 0$ and $c_1 (i_* \oh_L ) = 0$. Furthermore, by this question, $c_2(i_* \oh_L) = L$ (using standard short exact sequence with ideal sheaf of $L$). So really, the only unknown Chern class is the third one.
For the right hand side, I'm not really sure how to interpret what $i_* $ does to the $1+2P$ inside it - I assumed that it means we now consider $P$ as a point in $X$, so the left hand side would be $i_*(1+2P) = 1+2P $ where now $P$ is a point class in $X$ - is this correct?
Putting what we have together, we get (I'm letting $c_2 := c_2(i_* \oh_L)$ and $c_3 := c_3(i_* \oh_L)$ here):
\begin{align} ( -L + \frac{1}{2} c_3 )( 1 + \frac{1}{2} H + \frac{1}{12}(H^2 + c_2(X)) + P) = 1 + 2P \end{align} but this seems wrong, because this would mean that $L=0$ which is a contradiction, and also that $c_3=5P$ which doesn't match up. Thank you for any help!