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According to New zero free regions for the derivatives of the Riemann zeta function assuming the Riemann Hypothesis, $\zeta^{(k)}(s)$ has at most a finite number of non-real zeros with $\operatorname{Re}(s) < \frac12$ , for $k \geq 1$.

For $k \leq 3$ there are no zeros $0 \leq \operatorname{Re}(s) < \frac12$ (assuming RH).

Are there known non-real zeros for $k>3$ and $\operatorname{Re}(s) < \frac12$ and even $0 \leq \operatorname{Re}(s) < \frac12$?

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Mathematica claims that the 5th derivative of the Riemann zeta function has a zero at approximately

0.2876 + 4.6944 i.

I don't think it should be too hard to resolve the case of the next several derivatives. The techniques in the paper referenced by you or by Micah Milinovich should let you find an explicit upper bound on the imaginary part of the nonreal zeros of $\zeta^{(k)}(s)$ in the left half-plane. Then it is just a numerical calculation to find all the zeros below that bound.

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Here are some more zeros of $\zeta^{(k)}(s)$ with $\operatorname{Re}(s) < \frac12$ found with sage:

k  zero
2, -0.35508433021047637343 + 3.5908393243989674267 i
3, -2.1101457926534055013 + 2.5842247720404084808 i
4, -3.2402527145532669326 + 1.6896108197133965315 i
4, -0.83754471075524375517 + 3.8476752430859881333 i
5, -4.2739256609673291219 + 0.66239093925605351714 i
5, -4.2739256609673291219 - 0.66239093925605351714 i
5, -2.1841011922856061145 + 3.0795001018135066045 i
5, 0.28760642074073336717 + 4.6944346849390517355 i
6, -3.1693828756887856638 + 2.289409321602478671 i
6, -1.2725578081548747958 + 4.0741784672743930742 i
7, -3.8750437984692690102 + 1.4917785168218934496 i
7, -2.3934461266307994251 + 3.4062662320064732901 i
7, -0.41331687971848942853 + 4.8452970349839958557 i
8, -4.5682125648555486581 + 0.81115476881064826173 i
8, -3.2523204085919068777 + 2.7169913803460029554 i
8, -1.6702731549827933708 + 4.2784414287490737296 i
8, 0.41829665603240684742 + 5.4752676150632522598 i
9, -3.945849046353689844 + 2.0451787413259943742 i
9, -2.6409589700825760293 + 3.6749136820039963426 i
9, -0.96722009647711882409 + 4.9985369336041978752 i
10, -4.5121442160084114922 + 1.3320691841545877923 i
10, -3.4228591767532529916 + 3.0609427521799546739 i
10, -2.0391435059208024397 + 4.4684432318278930578 i
10, -0.27483240240270603573 + 5.6133090845225833824 i
11, -5.0309783500584905982 + 0.76405382608151195607 i
11, -4.0769374324171281166 + 2.4384470640345412807 i
11, -2.9061882184131317446 + 3.9131827555009648537 i
11, -1.4413185933633706999 + 5.1493402952351168812 i
11, 0.41063992065310427805 + 6.1502251675161887368 i
12, -4.6217542104762414501 + 1.8307305267522831431 i
12, -3.6306820396013356257 + 3.3459003484685151686 i
12, -2.387362966467342712 + 4.6486063663821024758 i
12, -0.84520925190624343306 + 5.7472734659279260752 i
13, -5.1019135182813204449 + 1.1817501251741196135 i
13, -4.2445336021969936603 + 2.7739661902545951775 i
13, -3.1788394218248838054 + 4.1282947333700902908 i
13, -1.8653042754449209073 + 5.2971387830938879447 i
13, -0.24996831630193368259 + 6.2810810441530943019 i
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