NumberDB$^\beta$ Advanced search Tags Tables Help Login
Hardy Z-function derivatives at Riemann zeta zeros
edit · history · discussion · files · short url · special values L-function number theory
Numbers
$n$
$k$ 
$D_{n,k}$
1
1:
0.7931604333565061160138975652743521141458929050542895115057356964340492301490156808765812398148455407
1
2:
0.1303291553730543700564420223806970258648905741878163163476271509439693017585738890090021135699597093
1
3:
-0.1242465870259772145131239621453312645868848931031586248329570246276364668715635949663650833540976157
2
1:
-1.136839106827974822136114506620184792135343776935371519566934923723272728885676113342313932091858158
2
2:
0.1200110212624434731346505884311983468444816505709686169441604600753959857529813135136203439301243460
2
3:
0.4506866051842963603704804363491088516232805014471126765526075002514049673687452563661821609339028317
3
1:
1.371721287216129937306915288826670864859834715906966129966326468688205372870180281370358263092643033
3
2:
0.3556361351834682714798463844489337748653627067031689711114322311609717006743250982420937583665312490
3
3:
-0.5982279291169028407200304740791688853994738987379873967200107558661140218584683364207462351187867827
4
1:
-1.303940950403577164427204160751025526096379113255019128934907045568021643730757326001351480394769124
4
2:
0.5933501024036607387676879871339079858507201399100216540800662814040166341515110666644703440176898226
4
3:
0.8772330784159287437057852248107484511016592282609615596733385523181121291077678851758652164755988758
5
1:
1.382119536855625937259458512500830837927579235407658693791256201180095840876098134687606468327794963
5
2:
0.7375918247093515360259117665921338810340978308005605196169012022643813913785859926170771043201137030
5
3:
-0.9216778289466460877312025634163241890725506410836278569253634843832800733204274633075168178201220744
6
1:
-1.936507965292594320826139328228459396395556890975793150427620646402877525424298108112021770470322560
6
2:
0.3865963704472802084942779495625308449235639896177211550492068009352548174688659430599345852902651760
6
3:
1.394080934839470813179119622124351290246932955392714925913927139571184847774168125257422718655066390
7
1:
1.490610761488820507336696096188517069741021482042700702817101013135043371646169215724296225004782598
7
2:
-0.3423112634145936504242296217285879763503201547114028950111525834814160539454881690467266543413894147
7
3:
-1.602446441748589681479524014624179372437490738936767334627139546193253294370879616889806947174241210
8
1:
-1.833511126245459185640716372703073621258343041141680840755131365802217072879871046882641022973097894
8
2:
-1.084226727103666690156571697769591589925335077321077660564871807297659632137692537160670330983932357
8
3:
1.427653003784130450193426075514701668758406887877764144800218262303050162964492620193052774253602433
9
1:
1.568031475625159112262396365129675709338833402820611513731675187160508188246471637912239133396615189
9
2:
-1.409771982667048669195718380348124728831118258819359988469681310292185303308135017931195594749645772
9
3:
-1.447722187793984439615698940012967145962066957656632413263462051508869206195079887133977478788194625
10
1:
-1.418932250429310962803907609211528641667986613949396727545534042451336401773526775158161377140644253
10
2:
-0.9222763887141006802094958120350864087377630174183483312537744846943003986363201934255117638710820098
10
3:
1.895172967026444778733522293021875368696133939806395843696697595124354247972280299300150149940999243
11
1:
2.426579067846444330335447677644931622551917311630154839538381128382907915417266712287478014792598373
11
2:
0.4801940624608764056092115437687150296157643088115912910223148418483828510838338269264615650963880134
11
3:
-2.390473501662485706070461764579693869928402877022535014730798834578781533014478561094000820631559037
12
1:
-2.367934929295201019319303584562341126747025148391684979322854453852296591985958578486804209787874479
12
2:
0.6199706534704612677848742269863098421179291638028817990625337936681067890188039758476758744693009936
12
3:
2.529610168233767329833189289117635374443195187291214989708479664506045048587392987625151396395420783
13
1:
1.391805616760924052785106130734913366579480032082315730189413132226231712738508269966831663032352086
13
2:
-1.030009771236067116929725095027988002415623619203902807550031010951605907116542287939136134449947908
13
3:
-2.473142380387552855629897417137768148152199968145970421383428546764449376745194418339773218408937681
14
1:
-1.654148199526990772892295031701829142389994168022142155039001025130129800356999958768833394644850196
14
2:
-2.024634906664796910541779978315669655810231645888289305066387196488357750849790188535212880666517831
14
3:
1.563848911975268032668037282319886159460262855175083638318303313255523142833032677550515186096783275
15
1:
2.287779014022755267395404337598252900512269602462133914791808505824526385292932773976772307984225168
15
2:
-1.742918884671902620981488357212250784435442916233584739353734293105688614612082388619758715039163528
15
3:
-2.478458242804134818537703049933796599460447079532712161706452459828723714138854620925322795845029158
16
1:
-1.783579559212896987719466069378529926738382010804676667385999988563276312149670209361865124037692447
16
2:
-0.3533452551504895367803945443790621134840279946667921367982576720231199722605330465954411618267308285
16
3:
3.162286134543624247252737714266518163192597441098584710475307515468268138957065174697361859271759514
17
1:
2.186311017564523721704765167151056669671347625946196900538805024179114852732910307690145413471461416
17
2:
0.4174236070611453170412329559961861143924515526509839121282122219492887542475397135792893337263122474
17
3:
-3.313668225850410272890785876655765709052925510179047954236494646102906106184576121162123186916583224
18
1:
-2.964533986378066153041618503436010471449137088272701708862012791975719676409853006963364688745267045
18
2:
-1.039508938311016654116041221422808079285157566544112111124621030848113050024393865772492555811598086
18
3:
3.599793459802133054097635412667993585652836266823903771205947265631030046134443583347219552702379689
19
1:
1.779555995309000613722624589442353030249744745369628324227312813153336732050460971473598514297090492
19
2:
-2.156901747157178776765311260371754821785281026808396972099471828635061860228255924740182056764311167
19
3:
-2.223918313009373633430139247495291862522685909537287831398016419882000023084903238081944045679109703
20
1:
-1.457542590500210262100127083393039125937323199505030930631778916615570276728402913132927293154403633
20
2:
-0.8919397374539004235825746641483060171332331330849919672002437458371717320401722772178340505692616857
20
3:
3.459293862831141851885538814267587765206311747172902665126584204786659750996487888414846040488694780
21
1:
2.637886209574003391674753225607593813259214790420252372963470156924311725207124090557341350259353623
21
2:
1.819140106920633857768632093824394804186597579897949399080677406037590723492881373000261546910256695
21
3:
-3.207696099200225867385831953164870168703424682884314496623840842512411820816282970568286132508103995
22
1:
-2.626873559215977239601799133059937037938469725799170510499086331635079632455759104314830673693727493
22
2:
1.992302043474125967257942488200286730888671386478231816663099857895214921568904924040561058796680234
22
3:
3.596737416629257565830242644406886314879743537754161982989709221336325776364240748816183041185829313
23
1:
2.161778835136019920864836019148068845336076164088077158798844853656650867996153022251417284313964089
23
2:
0.6348409026894252841276081121749234202995333391779512596220896471977331167538964884690169188513709760
23
3:
-4.179235744617845072484862370471544993931909866581040086232086978520588025032129694074098162366988209
24
1:
-1.800443497039863411379764332251133445220710949576889422649879778915531771597068687331978461189168635
24
2:
1.281041682647194675404381858505329435444683947214331859924378772853935716403063964097501937572775185
24
3:
4.096150179587753402659175750069507773479721936777300842536952487065115807811205260894421190151340311
25
1:
2.176460781293763868972578079719487753232712762756320129378978966304108140170060158721650330425094166
25
2:
2.790571540015701592362027160816032094201334005295740434976758160454488778157895284601154404910739003
25
3:
-2.763557817351947457555498184464186883931832168159095940188369365468270085880143600320913728669014396
26
1:
-3.005083384109537333009683580613427912411047958193213064022235805335269451208279122678411791919862563
26
2:
2.149716486910415553893097985986566275993845113325444901697034126325888548465549333871365278034654311
26
3:
3.904074094922711867614942778023463508910355118664974756846144725952693622744048666613215047962788533
27
1:
1.479402182945277396420359177000549712189635124177214797489571716680615404956086864040978532938413267
27
2:
-1.367977793946788320061923276794862755746430572886697037542035944202008713528457590051546039496280238
27
3:
-4.028244437282722571592860882940259246472104870863446338954831418283884919915054675204843105155985588
28
1:
-1.693559744529100910658063493476115298653307546332759324266591037975522271347074925530850521409197250
28
2:
-2.361462776239707860254286486599730725576182711410146923420433389974984933380353851138870707590235198
28
3:
2.882307765270985961486729998230236349665534312592163766837337252140397844454532424699003786437269784
29
1:
3.515767073555275391624589895262019940248967653770380979097497439987251382982199561932828673410235942
29
2:
-0.1225717360977535951505863678587442034315241216627904660769188668049886696548868917331746582650490667
29
3:
-5.604488194981454523846699222612423434111175498768180487268783084137116413709179854023963912742231584
30
1:
-3.154647299576244776388101446096408979864004094280393673038348675058316147113303240527257759982923499
30
2:
0.5712869356798284231088680590804149985881357899402340885107818270597061987638257803574020714953988783
30
3:
5.494054906981110667663260836971591611744388246511784253735030860676453773828448360023407176656328155
31
1:
2.167414573753041126042542174715524519420841930723536492003537377176173467085724712119679046723671341
31
2:
-1.251048377649978092905036706892874744484292706954506220164795127166050149563336453004423519193562946
31
3:
-4.578974221867956136855661702520485595108810000212566403384042545914658784554575729340804226796088767
32
1:
-1.873685476057484158575871617950842230608267648336926342595622257475573105499145091323691900497281301
32
2:
-0.3154069128240278186189071142250514802361190560886166270655358349420598112034276814908239821418817971
32
3:
5.168785822524009207616597590989022622222031271430793234091467580114823989418520618304835758421741036
33
1:
2.982497154120917293746750696419562203185308424296130473377522563814092387419864472085679833478547879
33
2:
3.070253844353233247710759594094144007526328090410207770655368254295634951807720985948826751107775619
33
3:
-3.882434027619610506048404979348553684333363097017086657066487809043686648633914857197598101754479889
34
1:
-1.590845759488853695719282460705032346161634322756142259939026523879951933577752313876905074270875669
34
2:
3.840126997112683656112353135615741403480211894221644054848373596460117235883643691138474334378188572
34
3:
1.483430871351775120455123128602778407834639662275078364718263654107281690450135128571347569616936233
35
1:
1.361150776893148414467903241774149370889556675012285583864256572277825238030299043411425040676080853
35
2:
2.257488593003051110307076273525229687890127813167086666232110860350030255293736597011283603876455685
35
3:
-4.665716176445615168457744143435717968211996712462344383137780968248878998283609673908179536132616407
36
1:
-2.546278408985331523750513199031959539813706760056104830075331493605442615319863446585022356736324100
36
2:
0.1099872196422878262621077841177475588745237298193792742740746110277417493456668916448463665229018109
36
3:
5.699853165865148381820945418281442683004495453595786482643588954943396071036457966640372799385263255
37
1:
3.119005986211206877541698891647744378626609341298370492694660659374242823163228743410652708197239067
37
2:
1.549347229326353299224729921678047865799231333189765261850594336476733256567201917259941748893454454
37
3:
-5.969749163327410219908283203861676610739369936987703125051633256776936603475619464761197870019686109
38
1:
-3.887616998927383771529894196059293747061550851031962232303933561875535077191317836841899491536149135
38
2:
0.2968454428542946438975061175894747469762119703331623023174473284543368300567382309043612679984263053
38
3:
6.867754347494864904743277385032244115796762713571643108045982586387427489023062354175948050650352756
39
1:
2.294939520333356430696215475642475047964996573173424792014746667923381986856159844796321230023475624
39
2:
-2.174980618142228794522434816922563673795719442215070896237808318059080332507606490462373536331197765
39
3:
-4.536236833591324209881659108092148658208731221075967717013482790303888907871086622683086466266688718
40
1:
-1.559145598667757245168750936095521104578797247070062301391571009247015316080506480115673072054557509
40
2:
0.3225386125274594377179961725906995534772592960978103821771214795302502956508931758467636758659382703
40
3:
5.861105294886790687958890287446357007931257069317114332498301440297246287729636541364399375474159902
41
1:
2.315936363075534792375198006410377348163794518319988055146853254741543208291055797612678427135069069
41
2:
3.509350921170606024803595783083428830506969827424627816039881145705369717349810506506486161299434572
41
3:
-3.065508724134804990358719086016716340361860442062224051664986804486749030582095944019731297948563603
42
1:
-3.938417778543752380857538432797483960892255891390065582539031121432200297648560286234575439987363167
42
2:
2.284064136360684564417076120249131357686693574833856023956946916373897875031382541926518582954800538
42
3:
6.598781875344355085071828993386994448117485812224800471112416012043518557175159197956920592101434865
43
1:
2.343943244428389943892952114653066092603277527595685766496373980531852108290478097071083176802284594
43
2:
-1.446392140690924855833385415074032461484914398750776840088579654998715775711114913611526293505527543
43
3:
-6.125878898788469423699328069644648144181886947129616081824401249140920657384337316086879095657869442
44
1:
-2.319646831720941073892654601487650040105479263341160988515727763677114242793014178916810332554287944
44
2:
-1.394416382238116694722803888462214459833749302608850282715345511463401099271866130668426763193106014
44
3:
5.910210176849824505159031974569202884702762094932171563979886103141420108372052617596182656193869599
45
1:
2.289266424291143599589141144375502052786450129466788606981348484122822853123608279782486010124675147
45
2:
-1.639131894024754625005920816978347563578491034037395088803456567615399148270509009226102312193028358
45
3:
-6.698989641766075658175753572470703712121265853949625843769659587226388586469689390773160463067862178
46
1:
-2.817532373030913836413170553525238374619663390464509237288497437248950514932370286736247921027506950
46
2:
-3.955642816697699351492261119312341188926146687226470255388650928446876277782477809691829079682808444
46
3:
4.572102602377323074873040126126917649299586905894854859495297502547575500759502665863642041352484621
47
1:
3.159787098794175573727471808914870762820744480941624802079252192894253302930977518670112337346464965
47
2:
-3.558986791088667779692526392056921353756673465496348051050930768932591836245068727264760733096354030
47
3:
-4.815748386578089737167369931276546415634842357701436491235675537120426524592266506643747810987556471
48
1:
-1.801498476844353671961763435206061401515216237675099633648040179702251310291889102624841560978751837
48
2:
0.8088424339523664732135685516798573675497542184923869667406823647701409096255578478538601078734438812
48
3:
6.312871193151491615241480273772195355137815921160636841355151411379671018147274478714592271828438856
49
1:
2.018147575840671144790890749724333799690806431110997623184544457589437223296256705034273044190345901
49
2:
1.653749810889178698050861979987732032216723637647532951018969198903340912942364064486981561941778018
49
3:
-5.731852116095244858163846261845069288520985298133555449713003860518090797732937164762218610079820726
50
1:
-3.749757824965203453877240827781776264575246589579110428060679228234695531469544959602547364055412988
50
2:
-2.441000437717292724729420656036925770575266743407427618840219797888152879284327048335740994675038887
50
3:
6.864715518361038501497430501638806369751933817853469209868599670084854054974972475079019340182594000
51
1:
3.247935681102870393498643515421484063279379293201457327007442208552675520094340335955712021316326566
51
2:
-3.239542124666292976906242500166410509225985194727891958572078960021599782455658215528827853120939803
51
3:
-6.635534848083756855239476729356869823799076277412055024479623443243600742027214687380221754684158795
52
1:
-2.817195995525494566180437563994890835431089139932966564871885602892640492982036867946175687944384177
52
2:
-1.624761876778605360457418540072301059774634970251498394795919783097108456258189344180521484950908882
52
3:
7.533199248389550863341444777236418019013537581485584373630764723984052479332162544942426541141959988
53
1:
1.620164189462986600577721687506652293450360164082945803750596028992100200773197076113713009385534719
53
2:
-3.083802138441460655461847337271349788982257940582919957260573047924408353112607772784305590994679625
53
3:
-4.484280570532883772250640266975488899955076255231377164477394210971447096769251910006138295438797544
54
1:
-1.582216990833738082841324331218529017910748449671817350211264505862642584961821720039051426343440886
54
2:
-2.820615072934435142554359332531863111033032368452424963786811550156759523176181535596410751218476027
54
3:
5.109719454459362633733009635229682141693814651202723518149098084008665483466304388959432056497956115
55
1:
4.241970061210363175832061470343042794227147349930016211984491597813936583372639649104369725961948700
55
2:
2.988505005938153116578091923032948965220040005268816570699905544453029276023023986639373535001420175
55
3:
-7.365557569484537609454439808679340405055771622588207077223278317479395665054192800037004307195658375
56
1:
-3.095738538526665724662747995192601940076727558754852459404110400833132706874754378495629658234619288
56
2:
4.015959735965857851014560903824603422853507754201202305907428749461166905774694277074908463480817784
56
3:
4.872818868690310740593524387328652484834214628923414396764674540227886240279958316488805893570545089
57
1:
1.743129639939633353595341735834615317563653671444969632028171105124855323164864131236288112311398943
57
2:
-0.7996753284742136083923642111410508141262254410906699534073684716233378751451026449174230535625031002
57
3:
-7.166503953268474747893633991729804235765994092044771387800258262452602847747874367429290676882051215
58
1:
-2.135779999367715120159195900599579421600685715795534126018764224838429583166681512443618829533744700
58
2:
-2.536405890936252337156187911048398219596207847735965103587177313363032192413332336078531518091230502
58
3:
5.509705376216923525235134139413123236252374115109144577912345873004796781115458904476246952045246197
59
1:
3.590702840290744905125777398076661561099470866492645954189795327395897955385211656536809965725991460
59
2:
-0.2266814459618099516927795984950895354998375185046861928092559776869973001805473062953536232604811650
59
3:
-8.876857684567842533139089758657930135227904168355556335505236949119382189873977134076066457954154382
60
1:
-4.114423277641728355271784742964700104110443153201428819696668023815992551552017207262269051033485829
60
2:
-1.561123029264150201787897451435627013764192129279914295626195148308716393754264165164456359508157389
60
3:
9.292069585935869247773140064666515688154743194707265725797526934974258173209449767296440246819865988
61
1:
3.394306695494140729065411694876104121574520791107249067238614895661369574324160546375984160689162449
61
2:
-2.561319683115934290118896611550934353410241682188562829321154446207552405151899192000517054420289101
61
3:
-7.588039011931665045385961644828961221910898706805847423352823326648643667803434803731159622297303505
62
1:
-2.456068910603762704001534605640082525601425007592759407174531539205375826229405041788953795280990074
62
2:
0.2686395374314968734297241224999187507719685896179243297601792891219776815419227292042673396280490662
62
3:
7.661006203232616401084546642520899262939976202783052519939986326105323575946080319164840611288479980
63
1:
1.536075687484538613416279208068831860388631594263083235312924648290774153750687180328581532953963232
63
2:
-2.100749742565196203231759534210143813657143208280901475599724111674050620803692772449244016260959567
63
3:
-7.604983243282325293853525504829248804575084386122509246685630255008029304175955614076010311400955567
64
1:
-1.978937468126691391741332863067525950761198930634460912776736076151026138524413405627992268972425609
64
2:
-5.227090463144202657455069587836353384078546134139053989927593721358971027705538574698139719284380402
64
3:
1.453899638662072062533001113795092047182817574340573902322030820789250117989979901244858361216153605
65
1:
3.486440628243157740764648757441656099870834211399987097659077926674449767073353238682830995757925143
65
2:
-4.857798219009755871538002158276595549673531343914343815370530610243723887223820100456828529222832956
65
3:
-5.676809728262821248717758039249020589875376220970122573267738838803574478561720655745059751053934706
66
1:
-2.354940176796607103656156712116139356005290386237584391595576917346360985805562860876253978065560209
66
2:
-0.3081377024313666278048676354641977670321392815507438506437419868632250619943422817570224635664834774
66
3:
8.819972500780992134622614091251973242106563269476590709073802157877508562294298753410352666163360424
67
1:
2.658960802255405521688200492071885771747358632241159875442991534701257991660710213007662981885950675
67
2:
0.7029631615675439816122954858562515619280964030715174721467136658545259727926477097546699783449954958
67
3:
-8.019231718355724416773166961154496841828275780245805372578386106770835835508216097195952620818056167
68
1:
-2.820578744145183132975793634925776190175027031162107134027520473007240548590424995006074937848519368
68
2:
0.4489079918470185057999405534145398159416124094646571406272863383009343316413181300053098498560370108
68
3:
8.900898685156463948261685780106956213834970434370521815653966230045101576629655845567514528175860905
69
1:
3.506206017375450659669976361678848496389355878272475253948913516312166176095637315013335579606888731
69
2:
2.684839565456988380002797012676362065406864809268305981327763059100379097133006980228447988022274732
69
3:
-8.677739343756298171007635742610495997922702259831495057455139719586404532620843620577148767615946994
70
1:
-5.010613528798497734698597147254906505773280301433115519851620998762106320335247270662443382258401815
70
2:
-0.07515803733945737190831513438479195217974235093342868853778282983823638351066928071933329126242337314
70
3:
10.82473493238734261932737189981123662572904948589268428280605161631606183915486565402734224335696300
71
1:
1.494336966523860363573758283326152020473047189136664752060887100628754737623283419671351550363469370
71
2:
-4.239113863684108368618404144936162206859752066037011554528790511374481630685586245855831169867274058
71
3:
-2.208320238157328329799415419289716061355359131834701060382708746393829476460029927305267212683351452
72
1:
-1.233177031544400426763074053985145578837547130044725661028736946350733807606665064132594197713702922
72
2:
-2.145471031710974287006610687461228277702479658677629320031442779986756004820242392316665396635913030
72
3:
7.037618385555495384254375640743911466215289624567973708379144173679245245128694842740710924499275690
73
1:
2.835725439800282029792113297324855889082353984369330744176019704844928220922357466953392388111993507
73
2:
2.739780435722245405737814762098199109374381679754470039192158905449009963362296864320887710145674771
73
3:
-6.883204002810222358618293011054513025459690288682288463081800822090482678261511383889446999305767795
74
1:
-5.046888326832377741219807679149842674920642479554385424654101015379899521156126113711350144977353694
74
2:
-1.385884497548208976586299189272038634127894263814871191253490560533944628208553789612331477010698317
74
3:
10.98211134888637003539509150493852134045530539256049969837274908409980062261148512900808223660987826
75
1:
2.769652371396630755895252361371320930377440913169236766944395211978309502993561430574532654587540586
75
2:
-4.490731115488898550814157806783103771082741326794406465313183677666682292642619249641670193016265845
75
3:
-6.430151993864871299060520298707681621061173458345796712254318042893352664830796192414729678408522796
76
1:
-2.390647547894142527613741966050680093684956766709450324567308366070182584493060082742781382590962292
76
2:
-2.490488071684939829603706211985025518365834415320301073650312688841548383352029164884211474465986183
76
3:
8.755521192460982159527301619770762996262890715988753135956613989562981509402998964800798215948078660
77
1:
2.850131333090212789914835592981822143803671811687486429151822721118387714654188994805005322897367800
77
2:
-1.394571048259721371480343116608956016060139488259811671432522487233058465902293125846330419501455854
77
3:
-8.445831476580639092518332325012982958860085238212360414347882094432592858443781286972049164860097639
78
1:
-2.181056548204643597579082239018458044471149102793240047898923345719626421527241252313053870569803611
78
2:
0.8429211508530230278833815577978828092670189530779878000226539324105335161212731184123910813935889709
78
3:
9.714792602771063400116641835757529556847215448787901992566326996742811517289099847249544908911173060
79
1:
3.157905155276447421633011663865017412293780069254720389315151884909927029964512162913350365474285028
79
2:
5.582513701603691173487911081792456848952616649640089022423874983737728515003611684569648103207365389
79
3:
-4.859392204549002037660407877590411012728685651589519629855756100111578814334169183034732697344525942
80
1:
-3.382620228232170430002262097287976699005233617947639225596425074857158511008236226523399836129177370
80
2:
5.459268286747033131404942534795437907167696511894225861772100370574156012947466924119991960637228459
80
3:
5.428789671324466411558283036962068765974347013892442644132801330413937697977152068009788271730217469
81
1:
2.233096431689364431387822068534604560529071551527427052075317541162852251074519930898616602265155072
81
2:
0.3856436758535633979997725815545290649987976124151717279493294131628220498966113711832582323028593217
81
3:
-9.451176268598032077361306085525370167608823906048340264568652335078729132731232521546200864031338751
82
1:
-2.076325756990852204649759214188392577271678007148672698055420654247920206926968577944297058074981152
82
2:
0.7566333153878146245564306876404654731302536951125433383629992812789555311582183494806852294576027662
82
3:
8.884559944146275252312208021697211958792670676092965273504414461388754671978155897016154593753295133
83
1:
2.800287465790075251316209998545036260492159105681170033705524540309844144847283986086549232795244767
83
2:
3.990614568722502329611555645460020678590522906461265513884925368519610747011591171352143903128408413
83
3:
-6.485092863953714058838828021986361552368373154739677577492961651236736317917118462526967222602549148
84
1:
-4.517097004874417380245123178927266939167189050646331957524452406444502344232932642393679335438567768
84
2:
1.947633881744920418005403117847812089751223876270409421790175079740158715148729427739369275865139392
84
3:
11.68908604580274423137232651146666682188675848523323923392077643905069753991621832069272943767065847
85
1:
4.117986512910688971162823870437756811564837548905164170008035972607143152606991156298316751389476073
85
2:
0.7460669359126820700294701261499375252169043175726023298752349848027462082156014079410598719905644801
85
3:
-11.82150049407646802564263712595400361629189845113855284034754513743157963092074491942088329185046010
86
1:
-3.310508559804980579921635484073243545303862774929643592583980562017892838317136995777134258914928740
86
2:
2.108771833304829053105640338636222466510071801933480364930714569538505399758428415994423023084450893
86
3:
9.233883814243987890625362535737165859545869958531080101112854134742410375241005723808826869541898407
87
1:
2.085222144104523646076300942504814490554876299419613319455057472198834885542732617426355367035218507
87
2:
-1.452242868534862557779009450084680034760626948130910086092742873031972772876152623345307664663920916
87
3:
-8.600935878844307607436033223784805585986409292691774502886261339912950800571341840811196427517128113
88
1:
-2.209365835607717405081347902664007205406418806707749243025201865683332398128498990383116867936302648
88
2:
-1.965718759726944057089940282585010744969600152619217997495919552532694028411096308562685997471847249
88
3:
8.604929586003011533035434113410013784580173793434360938799973052588245735753250070678568891799363790
89
1:
4.414511310304729689626432391727226883013497478801579502789358076018537673106058305093310417737561401
89
2:
4.752583880436158128959948110810549645468365698822949278334592254612496841278680168765936460421622883
89
3:
-8.609231118484885818013703903875177207406990706405087173037351517291301399672868896664134164906147410
90
1:
-4.128083396893856364146485154805669137896136478427683632325578054231953947492184421483990391709990172
90
2:
4.948777479249402248861619866381737692546588601957403340425735466931779741220901827158201387958508770
90
3:
7.627724011463435502286548003161061902801952159132137884788459793508496265278932788653605252057080521
91
1:
1.475987480670321756376654721741438916422518623007984808518732766891136337854370602864485922875257455
91
2:
-2.991874940413622351327454467176451726478013743314393489712185019439724973643397320885303241171976189
91
3:
-7.518441845577293609464672925258312368079882467915755446996394518768626255063634728684156745130327814
92
1:
-1.619105533559405885614609424613914906616895735116295372259962926999367714042541410324059510453353370
92
2:
-4.163767554430882999246574895382620554182516953913129192636332063155159657352166220954642903223580958
92
3:
4.627914925901931019444385626754515448174050850283129539326000455826534403266655398594016805935100431
93
1:
2.647722730520257510994154205748756552639195261572033597592526907398376950801591601741094503468933342
93
2:
-3.496313397971188062903966547771146810215205898064828441257662410822681017833524940308652104918500289
93
3:
-9.522886991802017823075320836525743666549260024502549229859890704363008431461979953017445564352351985
94
1:
-2.886983026955180748940153545682824217770494080689644752701888723623401447887468219486259374307128226
94
2:
-4.847919127562700460735023648688658103115624808393469762456696523146475942988042355383593586599416225
94
3:
7.938937522834234773558282386350284910095483394086308814693399482433777940904140901305812551652766677
95
1:
5.145022510260600332176295334988660769867449219630769114303156173344161840943078538591771539945757056
95
2:
-1.519606341499035755933819150493199195912080885092444852969601070252818730226843793532152192134929287
95
3:
-12.96790182623935127450915356583036246296154543396916655462976065738057219322873782757238552256515940
96
1:
-3.716188697275903389983539685011825228707442764718676385958917121316876210841071506127564273373333009
96
2:
1.744988267902518991707587582677940642848060699668088215465471532874472431677186231044130395244539210
96
3:
10.65957753247878709589437926251519679427837232377670758746357859972608929790022811166338107071125411
97
1:
1.478973502224087398897100403080016496196068991787507206620713924110801675734068213781641052083388769
97
2:
-3.234679583897349216435568863537796905530817652134124087066500687160082985088783680564734119455810748
97
3:
-6.230196818289141636998642402668612806470267691416532197226543035622328753395097964147173615064968881
98
1:
-1.466963340103393966983562436519612989617239326534270898779470822084619582601718167604530548904324027
98
2:
-3.130083951535719782485429842457594450407441724992488661608916424397527448553291603487101577171313561
98
3:
6.604518441761024141023022601813770617325627031188254272370496812392985279008011833684522156949665385
99
1:
4.318344467902397219341798978915952654557281954211634133064236144686485736650561447777835097404351481
99
2:
4.829225005445633253665235330696585038240063302269789775946894808096329976587419723426757708682860327
99
3:
-8.490629938823294823754891709671588958943815450440478273301151916799181631191177660263009258204514626
100
1:
-3.995024594193921608981685037857349532138898889424841141380958511686812768221562279735745079306948292
100
2:
5.499285147652382619815662541566811259142623670323628250251356935745775966743322423841433803294504920
100
3:
8.767665636328264132820686876264859323267786750418751799644297627481951099885453803499647124366691726
6709
1:
0.4145578923786792308103486338609959501611392179417716372491639181174023056077084514169834070443505384
6709
2:
-21.30280171014898548833510897907836347306550642431804408233707090854453182067225040983747342028073070
6709
3:
-57.15900822606084685671493474115650259667308304313433960471741535540432735516115346531704427488254674
6710
1:
-0.4270369340719351015087080077445555852871962632798090583326254142558794119696159071325155455756244724
6710
2:
-23.28819164653778447186052695313452754786405650759985070852862209208943777557557489554952833211779009
6710
3:
-47.97369545005228364997494000228648554556700273961353909589911144683919973718488368852840458865222031
Definition
The table holds $D_{n,k}=Z^{(k)}(\gamma_n)$, the signed derivatives with respect to the real height $t$ of the Hardy Z-function $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$, where $\gamma_n$ is its $n$th positive zero, counted with multiplicity from $n=1$ [1].
Parameters
$n$
—   positive zero index ($n\geq1$)
$k$
—   derivative order ($k\geq1$)
Formulas
(1)
Writing $\rho_n=1/2+i\gamma_n$, $D_{n,1}=i e^{i\theta(\gamma_n)}\zeta^{\prime}(\rho_n)$, and hence $|D_{n,1}|=|\zeta^{\prime}(\rho_n)|$ [1], §3.
(2)
$Z(-t)=Z(t)$ for real $t$, so $Z^{(k)}(-\gamma_n)=(-1)^kD_{n,k}$.
Comments
(3)
The phase is $\theta(t)=\operatorname{Im}\log\Gamma(1/4+it/2)-(t/2)\log\pi$, with the analytic log-gamma branch on the right half-plane that is real at $1/4$. This convention makes $Z(t)$ real for real $t$ and gives $\theta(0)=0$ [2]. In particular, $Z(0)=\zeta(1/2)$, whose value is in $\zeta(1/2)$. The function is also called the Riemann-Siegel Z-function.
(4)
The heights $\gamma_n$ are positive zeros on the critical line $\Re(s)=1/2$. For the selected indices, they have the same numbering as the nontrivial zeta-zero ordinates. Every nontrivial zero with $0<\Im(s)\leq7006$ is certified simple and on the critical line by the attached finite computation; no assumption of the Riemann hypothesis is needed for these entries.
(5)
The entries retain the signs of $Z^{(k)}(\gamma_n)$ and are derivatives with respect to $t$, not derivatives with respect to the complex variable $s$ and not Taylor coefficients $Z^{(k)}(\gamma_n)/k!$.
(6)
The indices $34,35$ form the first Lehmer pair, while $6709,6710$ form Lehmer’s historical pair [1], Table 1. The derivative orders $1,2,3$ supply the data used in that paper’s close-zero criterion, Theorem 1.
Programs
[3]
Python
import mpmath as mp
mp.mp.dps = 130

def hardy_derivative(n, k):
    gamma = mp.im(mp.zetazero(n))
    return mp.siegelz(gamma, derivative=k)

indices = list(range(1, 101)) + [6709, 6710]
for n in indices:
    for k in (1, 2, 3):
        print(n, k, mp.nstr(hardy_derivative(n, k), 100))
References
[1]
Aleksander Simonič, Lehmer pairs and derivatives of Hardy’s Z-function (2017), Theorem 1 and Table 1. (arXiv)
Links
Similar tables
Zeros_of_the_Riemann_zeta_function —   stores the zero ordinates at which these derivatives are evaluated
Values_of_the_derivative_of_the_Riemann_zeta_function_at_rational_numbers —   stores $\zeta^{\prime}(s)$ at rational arguments; the first-derivative formula relates that function to the height derivative of $Z$
Zeros_of_the_derivative_of_the_Riemann_zeta_function —   stores zeros of $\zeta^{\prime}(s)$, whereas this table evaluates derivatives of $Z$ at zeros of $\zeta(s)$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds derivatives of orders $1\leq k\leq3$ at zero indices $1\leq n\leq100$ and $n=6709,6710$; the prefix provides consecutive low-zero reference coverage, and the two higher indices are Lehmer’s historical close pair; the first three orders are the inputs to the close-zero criterion in [1], and the prefix cutoff is a noncanonical computational choice)
How they were obtained:

FLINT’s certified zeta-zero isolation and Hardy Z Taylor jets [2] are evaluated in ball arithmetic at 493 working bits. Each zero is computed afresh as an enclosure, and its entire uncertainty is propagated into the jet. The coefficient of degree $k$ is multiplied by $k!$, retaining the derivative’s sign. Every output is finite and retains at least 466 relative accuracy bits; each stored 100-significant-digit decimal interval contains its computed ball.

more

The finite indexing check isolates 6710 distinct critical-line roots in $0<t<7006$ at 96 bits and independently computes $N(7006)=6710$, where $N(T)$ counts all nontrivial zeros with $0<\Im(s)\leq T$, with multiplicity. This proves completeness, simplicity and the critical-line property in that finite range. For each of the 102 selected zeros, the total counts just before and after the root are also checked, and $Z^{\prime}$ excludes zero on the root enclosure.

All 306 values are recomputed at 559 working bits and compared with independent mpmath root and derivative calculations at 125 decimal digits [3]. Eighteen values, covering ordinary and close-pair zeros, also pass 155-digit mpmath checks. The parity relation and the first-derivative zeta identity are checked throughout the selected range. The first 100 zero ordinates agree with the stored zeta-zero ordinates, all 12 overlapping derivatives agree with [1], Table 1, and both close-pair diagnostics agree with that paper’s Table 2. Known nonzero and zero-count controls pass, and a deliberately perturbed value is rejected. The independent agreements supplement the rigorous enclosures; the mpmath program alone does not certify an error bound. The attached checks.py reproduces the numerical checks.



API Python package Skill for AI Privacy Legal notice Source
Data licensed CC BY-SA 4.0. No tracking, no analytics, essential cookies only.