On the Argument of the Lerch, Chowla--Selberg Formula and CM Values of $\eta (\tau )$

We give a completely explicit Lerch, Chowla–Selberg formula without absolute values of the Dedekind eta function, and give some applications, in particular to the argument of individual CM values of the eta function. We also give precise conjectures on these CM values and a large table supporting them.