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Zeros of the Hurwitz zeta function $\zeta(s,a)$
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Numbers
$a$
$n$ 
$\rho_n(a)$
1
1:
1/2 + i * 14.1347251417346937904572519836
comment: The imaginary part is $t_1$.
1
2:
1/2 + i * 21.0220396387715549926284795939
comment: The imaginary part is $t_2$.
1
3:
1/2 + i * 25.0108575801456887632137909926
comment: The imaginary part is $t_3$.
1
4:
1/2 + i * 30.4248761258595132103118975306
comment: The imaginary part is $t_4$.
1
5:
1/2 + i * 32.9350615877391896906623689641
comment: The imaginary part is $t_5$.
1
6:
1/2 + i * 37.5861781588256712572177634807
comment: The imaginary part is $t_6$.
1/2
1:
0 + i * 9.06472028365438761925536589143
comment: This zero is $2\pi\mathrm{i}/\log 2$, from the factor $2^s-1$.
1/2
2:
1/2 + i * 14.1347251417346937904572519836
comment: This zero comes from the factor $\zeta(s)$; its imaginary part is $t_1$.
1/2
3:
0 + i * 18.1294405673087752385107317829
comment: This zero is $2\pi\mathrm{i}2/\log 2$, from the factor $2^s-1$.
1/2
4:
1/2 + i * 21.0220396387715549926284795939
comment: This zero comes from the factor $\zeta(s)$; its imaginary part is $t_2$.
1/2
5:
1/2 + i * 25.0108575801456887632137909926
comment: This zero comes from the factor $\zeta(s)$; its imaginary part is $t_3$.
1/2
6:
0 + i * 27.1941608509631628577660976743
comment: This zero is $2\pi\mathrm{i}3/\log 2$, from the factor $2^s-1$.
1/2
7:
1/2 + i * 30.4248761258595132103118975306
comment: This zero comes from the factor $\zeta(s)$; its imaginary part is $t_4$.
1/2
8:
1/2 + i * 32.9350615877391896906623689641
comment: This zero comes from the factor $\zeta(s)$; its imaginary part is $t_5$.
1/2
9:
0 + i * 36.2588811346175504770214635657
comment: This zero is $2\pi\mathrm{i}4/\log 2$, from the factor $2^s-1$.
1/2
10:
1/2 + i * 37.5861781588256712572177634807
comment: This zero comes from the factor $\zeta(s)$; its imaginary part is $t_6$.
1/3
1:
-0.159433165603863091638468168180 + i * 7.18483506190509648215576440365
1/3
2:
0.342658233019985813410207892891 + i * 11.4313703381847467987254458482
1/3
3:
0.241816596008175157429658036252 + i * 15.1893510350981895095272263166
1/3
4:
-0.0368527585709669276640129965065 + i * 17.7687979773851213110711567918
1/3
5:
0.591799464010925724800913919127 + i * 20.6904413047432925490383841402
1/3
6:
0.193274858114286543588206371709 + i * 23.8978923530530417927546038688
1/3
7:
0.127980431714659812451180590840 + i * 25.7062991216611623352968391350
1/3
8:
0.334412678069944169401686796942 + i * 28.5249004246452971473762464326
1/3
9:
0.429118240867264720881997600815 + i * 30.6462725564512103990620263275
1/3
10:
0.462091196781564502295388168558 + i * 33.6434764151771296031068291174
1/3
11:
-0.147875412162700125763474775449 + i * 35.0086056161072210273743756266
1/3
12:
0.506387900981645385960264859037 + i * 37.5715137605346882219880192008
1/3
13:
0.472650578757928592304836022610 + i * 39.6960570839325771592373913684
2/3
1:
0.166871298806414106748555702339 + i * 10.8219336277245715759920505905
2/3
2:
0.570053320539946386911708343154 + i * 16.6058916606764683730182111183
2/3
3:
0.00261444020809895317369924833473 + i * 20.5251961742639433007116532372
2/3
4:
0.850926754557726141283189505528 + i * 24.3404084270052348009661321431
2/3
5:
-0.113771315918579559035260653661 + i * 28.0783120266226775019297041967
2/3
6:
0.721483751781628653777057851597 + i * 30.7921169159724692277929580720
2/3
7:
0.365783258531937213580312613449 + i * 34.1367093766771655983706352438
2/3
8:
0.460199615763710699795020958614 + i * 37.5838251515065394615629465931
2/3
9:
0.203973656808945423925683119177 + i * 39.1599753755528907942647964590
1/4
1:
-0.231398684547660149892818170802 + i * 6.16591066546094195072834120469
1/4
2:
0.231218444911129415582805346519 + i * 9.94458916136703117873757576037
1/4
3:
0.307983738069873093767824620521 + i * 13.3615873589608583536980265462
1/4
4:
-0.0946303762658115985593741976359 + i * 16.1174596758179864795092617381
1/4
5:
0.306240083107864296937888542901 + i * 18.3315569322345563380114162318
1/4
6:
0.521888203346763014248571591502 + i * 21.2789893483703143365049217151
1/4
7:
-0.0784692987847381880533211418243 + i * 23.6791591101364774345898299596
1/4
8:
0.318092718301595265677938144507 + i * 25.5783735553348396698653897052
1/4
9:
0.184955960931542901996767232836 + i * 27.9981961646480509111762554358
1/4
10:
0.488687339887289033876792641337 + i * 29.9807131301438250369376955642
1/4
11:
0.379207089958816493492599162598 + i * 32.7181515392452593473062231618
1/4
12:
-0.135682716495852163466183248217 + i * 34.0093428632625899902626711518
1/4
13:
0.396464024655020027566445154469 + i * 36.2599577929829775166621008784
1/4
14:
0.431669196825430876664968193713 + i * 38.1913369661064844249698067282
3/4
1:
0.250817018068039196989263125131 + i * 11.6706434065585016916588372838
3/4
2:
0.574311645486463806256177140933 + i * 17.7729076164685237412908738216
3/4
3:
0.0871556880508019157119552655603 + i * 21.6583855741676631700372438791
3/4
4:
0.905861230156364065449081548226 + i * 25.9590033141806956127058987741
3/4
5:
-0.153062854307726881947965283915 + i * 29.2990686750000498208123556875
3/4
6:
0.807896194300115783696150990066 + i * 32.6827536990281302182918813913
3/4
7:
0.437305762609698456518165517265 + i * 35.8257004751180327433953044860
3/4
8:
-0.0540283644119743239602839727561 + i * 39.1868476487502378759654496675
Definition
For rational $a$ with $0<a\leq1$, this table gives the non-real zeros $\rho$ of the Hurwitz zeta function $\zeta(s,a)$ with $\operatorname{Im}\rho>0$. For each fixed $a$, the zeros are ordered by increasing imaginary part, with ties broken by increasing real part, and counted from $n=1$.
Parameters
$a$
—   Hurwitz zeta parameter ($0<a\leq1$)
$n$
—   index of the positive-imaginary zero ($n\geq1$)
Formulas
(1)
$\zeta(s,1/2)=(2^s-1)\zeta(s)$.
(2)
$\zeta(s,a+1)=\zeta(s,a)-a^{-s}$.
Comments
(3)
The Hurwitz zeta function $\zeta(s,a)$ is the meromorphic continuation in $s$ of the series $\sum_{m=0}^\infty (m+a)^{-s}$, initially for $\operatorname{Re}s>1$ and $a>0$ [2].
(4)
For real $a$, $\zeta(\overline{s},a)=\overline{\zeta(s,a)}$, so non-real zeros occur in conjugate pairs. Only the zeros with positive imaginary part are listed.
(5)
Real zeros are not entries. The case $a=1$ has the trivial zeros at the negative even integers, and the real zeros for other $a$ are a separate one-real-variable question [1].
(6)
Spira proved that, for fixed $a$, the complex zeros of $\zeta(s,a)$ lie in a vertical strip [1].
(7)
At $a=1$, $\zeta(s,a)$ is the Riemann zeta function; write $t_n$ for the positive ordinate of its $n$th non-trivial zero. At $a=1/2$, $\zeta(s,1/2)=(2^s-1)\zeta(s)$, so the listed zeros come from the Riemann zeta zeros and the zeros $2\pi\mathrm{i}k/\log 2$ of $2^s-1$.
(8)
The table uses the standard Hurwitz range $0<a\leq1$. Arguments larger than $1$ are related by $\zeta(s,a+1)=\zeta(s,a)-a^{-s}$ and are not used as parameters here.
Programs
(P1)
Sage
import mpmath as mp
mp.mp.dps = 50
a = mp.mpf(1) / 3
guess = mp.mpc("0.5", "42")
rho = mp.findroot(lambda s: mp.zeta(s, a), guess)
print(rho)
References
[1]
Robert Spira, "Zeros of Hurwitz zeta functions", Math. Comp. 30 (1976), no. 136, 863-866. (doi) (zbMATH)
Links
Similar tables
Values of the Hurwitz zeta function at pairs of rational numbers —   stores values of the same function rather than its zeros
Zeros of the Riemann zeta function —   gives the ordinates of the $a=1$ special case
Zeros of the polygamma functions $\psi^{(n)}$ —   stores real zeros of functions built from Hurwitz zeta values in the other variable
Zeros of Dirichlet L-series —   stores zeros of finite linear combinations of rational-parameter Hurwitz zeta functions
Zeros of the Dedekind zeta functions of cubic fields —   stores zeros of another zeta function family
Data properties
Entries are of type: complex number
How well the digits are known: heuristic (agreement-checked)
Table is complete: no (it holds the non-real zeros with $0<\operatorname{Im}\rho\leq40$ for rational $a\in(0,1]$ in lowest terms with denominator at most $4$)
How they were obtained:

The zero search is not a certified enclosure method, so the table does not claim proven zeros. The generator first finds roots with mpmath at 50 and 65 decimal working digits and stores only digits that agree between the two runs.

more

The listed values were then checked by evaluating $\zeta(s,a)$ in Sage's arb-backed complex ball field, and after filling the stored centres were checked against the special cases $a=1$ and $a=1/2$.



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