As far as I can understand your question, There is not a direct mathematical relationship between CO2 and warming, although there is an approximate mathematical relationship between atmospheric CO2 concentration and radiative forcing which was established in 1998 by this study:
$${\displaystyle \Delta F=5.35\times \ln {C \over C_{0}}~~(\mathrm {W} ~\mathrm {m} ^{-2})\,}$$
Where C is the CO2 concentration in question and C0 is the baseline CO2 concentration (typically 278 ppm is used as this was the CO2 concentration in the atmosphere in 1750 before significant anthropogenic CO2 emissions).
Unfortunately, this is just an estimate, Earth's climate is extremely complex and the issue is that we still don't fully understand every variable and aspect of the system. The biggest of the uncertainties relates to climate sensitivity - this refers to the various earth processes that create feedback loops, which can both amplify and reduce changes in temperature. Without complete knowledge of all variables that influence the Earth's climate (and there are many) and how they all interact with each other, we cannot determine the precise mathematical relationship between CO2 and warming, we can only make estimations.