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Zbl 0557.10029
Apostol, Tom M.
Formulas for higher derivatives of the Riemann zeta function. (English)
[J] Math. Comput. 44, 223-232 (1985). ISSN 0025-5718; ISSN 1088-6842/e

The paper contains a new formula for $(-1)\sp k \zeta\sp{(k)}(1-s)$, which enables the author to determine explicitly the coefficients $a\sb{jkm}$ and $b\sb{jkm}$ of a previous formula of a similar kind due to {\it R. Spira} [J. Lond. Math. Soc. 40, 677-682 (1965; Zbl 0147.305)]. As a consequence, a closed form evaluation of $\zeta\sp{(k)}(0)$ is obtained. The results about $\zeta\sp{(k)}(0)$ contain the well known formulae when $k=1,2$, and appear to be new if $k\ge 3$. Numerical values are given for $k=1,...,18$.
[A.Perelli]

MSC 2000:: *11M06 Riemannian zeta-function and Dirichlet L-function

Keywords: Riemann zeta-function; formulae for higher derivatives; closed form evaluation; Numerical values

Citations: Zbl 0147.305

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