TITLE: Power series expansions of $L$-series
QUESTION [7 upvotes]: Let $\zeta_K(s)$ be the Dedekind zeta function for a number field $K$. We can understand the first non-vanishing coefficient of its Laurent series via the class number formula. Is anything known/conjectured about the next term?
On a related note, the BSD conjecture predicts the value of the first non-vanishing Taylor coefficient of the Hasse-Weil $L$-function of (say) an elliptic curve. Are there any conjectures about the coefficients after that?

REPLY [3 votes]: There is a related conjecture for the L-function attached to certain Galois representations, due to Colmez and stated in the beautiful survey paper Periods by Kontsevich and Zagier (see section 3.6). This survey paper refers to an Annals paper of Colmez: Périodes des variétés abéliennes à multiplication complexe (Jstor link), and to some papers of Hiroyuki Yoshida.
There is also a book by Hiroyuki Yoshida called Absolute CM-periods, where some of this might be discussed. 
I have no idea whether this is related or not to the Rössler-Maillot/Kudla story mentioned above in the comments.
The conjecture as stated by Kontsevich and Zagier is for a Galois representation sending complex conjugation to minus the identity matrix. This is a rather restrictive condition, and Kontsevich-Zagier also state (without explanation) that "in general, one does not expect any interesting number-theoretic property for subleading coefficients".