Dimensionless physical constants that used to be calculated experimentally but have a known closed form now What are some examples of dimensionless physical constants that historically could only be calculated to greater and greater precision through ever finer and more precise experiments until one day, a closed form (integral or sum for example) was discovered that allowed for calculation through pure math? I ask because I've never heard of such a thing happening, and certainly not for any extremely famous constants, but surely it has happened before.
As for why I believe this has happened before? We've had thousands of physicists working over the last few hundred years on various problems in physics, developing and defining all sorts of constants. Of those constants, some happened to be dimensionless, and of the dimensionless ones with a mathematical formula, it is conceivable that some were not recognized immediately and physicists set up experiments to obtain initial approximations. For example, if physicists for years and years tried to approximate some dimensionless physical constant through experiments, finding more and more digits through better and more precise equipment, only to later figure out the exact value is a simple integral, that would count for the purpose of my question. The list here is not exhaustive, but I would be glad to hear a case of something being on the list and later getting taken off because it was $\sqrt[5]{7\pi}$ or some infinite summation.
While $\pi, e$ might sound like promising examples, remember that the polygonal approximation method circa 250BC allowed for arbitrary precision given that you kept increasing the number of sides. If $\pi$ was used significantly in physics from antiquity to 250BC or physics was a serious discipline back then, I guess that would count. As for $e,$ the infinite series was discovered not long after the constant itself.
Clarification: Only constants that were already dimensionless from the start count. You can consider constants dimensionless with respect to the standard 7 SI units for example. Tricks such as "we set $\hbar = c = G = 1$ for simplicity..." do not count.
 A: I don't know if the following example would be satisfactory.
As you know, sound velocity depends on the (dimensionless) specific heat ratio, which was initially determined experimentally. Later an expression was found for this ratio (neglecting quantum effects), which only included the number of degrees of freedom of the molecule. For example, for air, whose molecules are diatomic, the ratio is 7/5.
A: The only example that comes to mind off the top of my head of a constant known experimentally first then calculated is Wein's constant, though it is not dimensionless.
Wien's law states that the wavelength of light emitted with the most intensity from a body glowing due its heat is inversly proportional to its temperature T:
$\lambda_{max} = \frac{b}{T}$
Where b is Wein's constant, with value 2898 μm⋅K. In actuality it is equal to $\frac{hc}{xk}$, where $h$ is Planck's constant, $c$ the speed of light, $k$ is Boltzmann's constant, and $x$ is the positve root of the equation $(x-5)e^x + 5 = 0$.
According to wikipedia, "Wien himself deduced this law theoretically in 1893, following Boltzmann’s thermodynamic reasoning. It had previously been observed, at least semi-quantitatively, by an American astronomer, Langley." who presumably would have experimentally measured the above calculated value without knowing its derivation.
