The Chowla-Selberg formula relates the eta function with values of the gamma function at rational numbers. The eta function appears, at least in the proofs I have seen, related to values of $L$-functions and its derivatives at $s=0$, via the Kronecker limit formula.
In the $p$-adic case, the Gross-Koblitz formula relates some Gauss sums with values of Morita's $p$-adic gamma function evaluated at some rational numbers. There are a lot of proofs (Gross-Koblitz, B. Dwork, R. Coleman, A. Robert,...).
It is often said that the Gross-Koblitz formula is the $p$-adic analogue of the Chowla-Selberg formula, but the only obvious relation that I see is that in both formulas appear products of complex/$p$-adic gamma functions.
Hence, my question is, how are the values of eta or $L$-functions which appear in the Chowla-Selberg formula related with the Gauss sums in Gross-Koblitz formula. In other words, how are the "other sides" in both formulas related.
Sorry if I'm missing something elementary.
