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Let $\zeta_F$ denote the Dedekind zeta function of a number field $F$.



What is known or conjectured about $\lambda_0$?


What is known for the zeta function of a function field?

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So I take this as an indication that there are no closed formulas for $\lambda_0$, thx. –  Marc Palm Feb 8 '12 at 11:35
As B R notes, Peter's $\gamma_K = \lambda_0$ and Ihara's $\gamma_K = \lambda_0/\lambda_{-1}$ (in last link). For a number field $F$, from $\zeta_F(s) = \lambda_{-1}/(s-1) + \lambda_0 + O(s-1)$ we get $\zeta_F'(s)/\zeta_F(s) = -1/(s-1) + \lambda_0/\lambda_{-1} + O(s-1)$. The constant terms in these Laurent expansions differ by the factor $\lambda_{-1}$, which is probably not 1 unless $F = \mathbf Q$. For $\mathbf Q$ we have $\zeta(s) = 1/(s-1) + \gamma + O(s-1)$ and $\zeta'(s)/\zeta(s) = -1/(s-1) + \gamma + O(s-1)$, which can be misleading for generalizations to other number fields. –  KConrad Jun 28 at 1:31


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