For example, the following equation can be used to generate a Z table, where the mean $\mu$ is 0 and the standard deviation $\sigma$ is 1.
$$ \frac{1 + \operatorname{erf}\left(\frac{z - \mu}{\sigma\sqrt{2}}\right)}{2} $$
Ideally I am looking for the respective cumulative distribution functions expressed in a way that is easy to convert to source code.
The analogous results for the given distributions can be found:
t: https://en.wikipedia.org/wiki/Student%27s_t-distribution
$\chi^2$: https://en.wikipedia.org/wiki/Chi-squared_distribution
F: https://en.wikipedia.org/wiki/F-distribution
What you want from those pages is the CDF (that is, Cumulative Density Function). The rest is a matter of determining what parameters to use.