Geodesic
On simple zeros of derivatives of the Riemann $\xi$-function
Izvestiya. Mathematics , Tome 70 (2006) no. 2, pp. 265-276.

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We get a lower bound for the number of simple zeros of the function $\xi^{(k)}(s)$ on the critical line, where $k\ge1$. 
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I. S. Rezvyakova. On simple zeros of derivatives of the Riemann $\xi$-function. Izvestiya. Mathematics , Tome 70 (2006) no. 2, pp. 265-276. http://geodesic.mathdoc.fr/item/IM2_2006_70_2_a2/

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