export – Creating a library of saved function values

I have a command of the form J[b,c,n] which outputs a list of complex numbers, obtained by using FindRoot. Some outputs take over 4 hours to compute and once computed I would like to be able to save the values obtained so if I switch computers or open a new notebook I can get the values without spending time computing again. I would also like, as I compute the function for different parameters, this file of outputs to automatically update, so it really is producing a library of saved outputs. This could then be distributed with the relevant notebook of code so other people using it can save time if they want to compute something previously computed.

I am aware of questions regarding DumpSave and ways to save outputs but I’m not sure if it can be used in the way I would like. Any pointers would be appreciated.