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Let $F/\mathbb{Q}$ be a number field. I'm interested in knowing if there are formulas for the values of the derivatives $\zeta_{F}^{(n)}(0)$ of the Dedekind zeta function of $F$ at zero.

Maybe if in the general case for an arbitrary number field there are no results, are there any results for particular types of number fields, like quadratic number fields or cyclotomic fields?

I would also appreciate any references you can provide.

Thank you for any help.

PS: I would also be interested if anything is known only for the first values, say for $n = 1, 2, 3$ or so.

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    $\begingroup$ As you might know, the order of vanishing at $s=0$ is $r_1+r_2−1 $ and the special value is related to class number and regulator (See, for instance, Neukirch's "Algebraic Number Theory", VII.5.11). Also, for abelian fields, you can factor the $\zeta$ function as product of $L$ functions, so for instance for real quadratic fields you have ``$\zeta_F(s)=\zeta(s)\sum_{m\geq 1}\big(\frac{d}{m}\big)m^{-s}$`, see Heilbronn's paper in Cassels and Frohlich's book "Algebraic Number Theory", end of Section 2. – Filippo Alberto Edoardo 0 secs ago $\endgroup$
    – Filippo Alberto Edoardo
    Jul 18, 2012 at 3:09

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The value at $s=0$ or $s=1$ is known by the Class number formula. Here are some answers for the first derivative: see this Dedekind Zeta function: behaviour at 1 for the value at $s=1$. Since for abelian $L$ functions, the functional equation is very wellknown it doesn't matter whether you look at $s=0$ or $s=1$. After I have studied the references, I was ready to believe that not much is known at least for the first derivative beyond estimates.

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$\zeta'(0)=-\frac12\log 2\pi$

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    $\begingroup$ Is there a motivic regulator razzle-dazzle "reason" for this yet? $\endgroup$
    – David Hansen
    Jul 18, 2012 at 15:30

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