How do you calculate the lift coefficient of an airfoil at zero angle of attack? Thin airfoil theory gives $C = C_o + 2\pi\alpha$,  where $C_o$ is the lift coefficient at $\alpha = 0$. However, I couldn't find any equation to calculate what $C_o$ is which must be some function of the airfoil shape. In other words how do you extend thin airfoil theory to cambered airfoils without having to use experimental data?

This is my own attempt, I made this airfoil model of the lift coefficient of the airfoil at zero angle of attack for a project I am working on. It's derived from a Joukowsky transform. it seems to work but how is $C_o$ actually calculated?



Because it is derived from the Joukowsky transform of the inviscid potential flow around a cylinder it's more accurate at high Reynolds numbers.
The solution and the approximate solution to zero-lift angle of attack from thin airfoil theory can be found in ESDU 98011. The derivation of an analytical solution can also be found in Anderson, Fundamentals of Aerodynamics. $C_{l_0}$ can then be found by multipling it with $2\pi$.

Edited to add classical thin-airfoil result:

$$C_{l_0}=2\int_0^\pi{\frac{dz}{dx}(cos\theta-1)d\theta}$$
$$x=\frac{1}{2}(1-cos\theta)$$

$x$ is the chord line coordinate from 0 to 1; $z$ is the mean camber line height at coordinate $x$, normalized to the chord length; $\frac{dz}{dx}$ is the curvature of the camber line.

Now you just have to numerically integrate to get the lift at zero incidence.