Much attention focuses on zeros of L-functions, but there are also some interesting questions about zeros of the derivatives of these functions.
In some cases we know more about zeros of the derivatives. For example, Levinson's method [58 #27837][ MR 84g:10070] can be used to show that for any there is an such that if then more than the zeros of are on the critical line, where is the Riemann -function.
In other cases we know almost nothing. For example, nobody has even made a plausible conjecture about the distribution of the zeros of the derivatives of the Riemann -function.
See the articles on derivatives of $\xi(s)$ and derivatives of $\zeta(s)$.
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