Values of the Hardy Z-function

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Numbers

$t$

$Z(t)$

-1.460354508809586812889499152515298012467229331012581490542886087825530529474500625276419375463356820

comment: This is $\zeta(1/2)$, fixing the sign normalization.

$g_{-1}$:

-1.531820673783700535363221099993164653432550753818873817929205366443896617123399653077601463179396808

$g_{0}$:

2.340181668496731751802589468916663667919312206976854219201417043061416458106862142403329897956073531

$g_{1}$:

-1.457427047874012250365884370244627965371459570929878855472840000926900128183151787503340091445822101

$g_{2}$:

2.845091238051921949764361213337435946253730938695014816844393711966142697720425124418749576608290199

$g_{3}$:

-0.9252646433153666415555725108310523282893988993346135397713105210755718159041514715053632322488451346

$g_{4}$:

2.938121538493740564591648932251781652142591614211189798353732599465966784094944565103924045164300165

$g_{5}$:

-1.786721752366213077944253321303249912565711917821293980729418509753175339688747324395853963398432694

$g_{6}$:

1.098756930045979027710953574157834850970179423899114358500111800680462829491683282712891655201695749

$g_{7}$:

-3.662902949119917103030153567245110534384089402368974461200661805582663721480021844435038877063872815

$g_{8}$:

0.6882923716918532375706207785110066448976997917562042274907265964544528731146819834932362044778121717

$g_{9}$:

-2.011213891517499932549224272449196554254512276174432858245159726770484614884338730756628097834616298

$g_{10}$:

2.912390450796176479909421253548238589859360384154089004457572803619401735483255121744839839945978455

$g_{11}$:

-1.758164905191713462986910087856208306642018702155376621116230378868843958139638756520634787138760757

$g_{12}$:

0.5385857937546013514047324847450402991116763043712613430950150273566034687290071537116819042046670022

$g_{13}$:

-4.164398758501065810495729404173278397792692075338190222393288735838294529899854203764399861083202102

$g_{14}$:

1.053877329013568028109294022878779826947998184120505904955027782867016072150352056538920983312595574

$g_{15}$:

-1.540058258275695462100639049328747173326830975867626393282134795273142345796073800601121337862396756

$g_{16}$:

1.952737280800121793223231822634536442126624986486862064945361060546016254614432281272590769220805510

$g_{17}$:

-3.614395009277827825835134144740359736899650932329895699317171098078041150200203880769100273690597441

$g_{18}$:

0.5654782085851993725437429706523956881608493620467874908924439809799423711325267602661471185000606453

$g_{19}$:

-1.227123733659340814523404816872297895166454419243946121378372945582217456723731856371111676880275867

$g_{20}$:

3.998197681746888413211346756503440305850806860382788450374976967624341815432220597247808951281050086

$g_{21}$:

-1.170021422217257750157525234806500354342264558998858745231814048080169869352613214807077565876604293

$g_{22}$:

1.949771390151838057327847468138672138349370486837128067721626890719839166523747347245530609672848183

$g_{23}$:

-0.6417238934650786613826608102757755454762050547185461379093070115950479157940101444284198585203671971

$g_{24}$:

4.475366957045480689536539099501090760559138153943194854922958286810405931966006135183728801215447367

$g_{25}$:

-1.305789328064806393334243715981450786726153490498333806031938191041861505808846747368320910782388685

$g_{26}$:

0.4915214633745276482405073173130131971914724773386646308487429808712288374031025379895723119015000234

$g_{27}$:

-2.852168538716479846843022212530861872748692431016290544579995946189986991784063996516945333323413571

$g_{28}$:

2.691414116537411999003197635454209943606311998530453810384462313216538923186000137781953345147598838

$g_{29}$:

-2.059895845440496795984412747198940375238597622771424336106466747439042124087870270897431585266399131

$g_{30}$:

1.016911876787354408189306031161300526593595087157460717343403802928261064092770925695006625597584598

$g_{31}$:

-0.9743968788233739602609669434465168612381724254024464329153671091925475373977897609458973548777667903

$g_{32}$:

5.187022821270778892180338549588693628806613479892681882965161363169864269139762208555203481846849527

$g_{33}$:

-0.1415823734960171871140167101195282769152420004005641175403728141687019009107557666200797383857897828

$g_{34}$:

1.781093074424639147461509972662133070065586759112036981454409616194198676941786098710851975303101283

$g_{35}$:

-1.557454672852061682759585932322813233010045551280306845807073356750440921025492176472449320429966170

$g_{36}$:

3.090465688178469991936480237161665859280205223543149599007674432535070833978315518936199030291016532

$g_{37}$:

-2.717786526864372272561546857242033037597946148440428805524265402949358938767086790973057752463294015

$g_{38}$:

0.8153919919258341464631929951529408486727017237826175027691232300131320079925608657568406296254281145

$g_{39}$:

-0.4670233021992401027028317786741285744831389638963943793244009487171438705968442541795659432942277845

$g_{40}$:

4.563417072203780717023930932152780683270561101262199857088430649456679111814035427635236037897184080

$g_{41}$:

-1.862563835942512830551722159284720271110087933513489064652118940938914330525196559811052025427383136

$g_{42}$:

0.9971805191649042184575525869600307711179402887053480125555057974451466577930597106511034004149325772

$g_{43}$:

-2.093810651474401057585663101392096529946897962050048607096251442430043402923924918657183584384143626

$g_{44}$:

0.7309134599856839511127356119398246771961720230139962832855482500130897203383333596492110744967624762

$g_{45}$:

-5.049747345160536637932404593991943570651904835143578067589735227791197852053290582682300003068302655

$g_{46}$:

0.8796250116750270469924821540356774990809010463839428421605088178027628907142848682158336032310059283

$g_{47}$:

-0.7523164428023249031631956330777309821186454113627047390621026411848523849765960760141511965003876141

$g_{48}$:

1.594517288906088230111519239553896934154173693114677888126450193281729632290144641294858561250559353

$g_{49}$:

-4.191570275058367636159638805279089331647430027411431777755781060207826978781083256480129006038404468

$g_{50}$:

1.132298244395848160466358608847153462286925324662848357709668921515801087763993725939042977071592171

$g_{51}$:

-2.485664655712361406784838266256539593353816638371843547552877822387160530307494060477161083810944142

$g_{52}$:

0.3658685829321773352172702565039499524247310822541729449749925328281834098548106359613474700837767842

$g_{53}$:

-1.710048487217534978159547211273967609569384911927316707383485877346341365249090605612406608787077040

$g_{54}$:

5.044736641005290168647958287731625302529635264720513288286444131761365476814409923255238729735475019

$g_{55}$:

-0.7109469559692842497744825780414045318418178594040478437524993163468137033886017462352633123690519931

$g_{56}$:

0.6608358128964443440924444899387877821271608446960122433703346346060661257316083519238314575072112226

$g_{57}$:

-2.365897326197067713040917417664329506274037623301924466833562408991623240220752238846450171409995599

$g_{58}$:

2.123573148704736037548812672233592486233246643042783006514871849866685587032318288453581766962608270

$g_{59}$:

-3.491383656232605989285334219000846211442843948816671644003199116812119695841124907309249895246610408

$g_{60}$:

1.302778848027867410951441993083804310749495566706916919418132663513214703428447623231288448500822847

$g_{61}$:

-1.328857815190650854400717389187573158559790304950085948289367351972023630576268493519926019827257047

$g_{62}$:

0.008188337025869567256045178515771354402557911040220078018492556359067159463459986299387240707611730018

$g_{63}$:

-5.970893190074646583091217704358137195708510757755867747357478708683296092980053354903478894087179057

$g_{64}$:

0.7964158119833868339931526966334226857158198478133611881471898986389918166667466491725028488845281756

$g_{65}$:

-1.322859012945665192972172008020829549428626982991392287029218164389051174056693873487092643182963793

$g_{66}$:

1.758817931695042448205731050104801411553000741412310546176813784852130296798023973181667229421553404

$g_{67}$:

-1.333662308156349381196588227224059612870005937010629372047984720890608520205039361405525528685367724

$g_{68}$:

3.358833539263379680520129301856673616503425448564614469570816446299145793455507818189830935649014114

$g_{69}$:

-3.349592118422207423709521260292287985049651481122385577012837204023017644875564536424221762723984194

$g_{70}$:

0.1070330178568896099767079262416280588388645719656951677909847394050358263417318587958812768765856825

$g_{71}$:

-0.8786392573849723296222462954775314623420499099289580135388346694786356418327774835646908600987197772

$g_{72}$:

2.742587049478224086159359208525027942336694122243019994798263979022084232469273386812471340404635086

$g_{73}$:

-4.065320038236068399845694530516984894954321495142735565523079272374475514309416116681262310828895713

$g_{74}$:

0.6241863274622238154462602872760149234040123428839643453969952339595477217151010482181182206670644956

$g_{75}$:

-2.176809297996623216014726585461138188723857530635825907242077136294445420802380490616191390051561094

$g_{76}$:

1.263062409512163956344952624593753519210181060410389397213044461632326033438820221320933564597639634

$g_{77}$:

-0.4852023293099956262385671782470837441394286023694987250184299910692529760136316161624102019319313670

$g_{78}$:

6.062567723548795999507568493911098654835676140382939113605990684861360739513600957184021626025347009

$g_{79}$:

-0.5578958265702726174648400029260659593659950977872292445727227205627051184714095116369904579399696976

$g_{80}$:

1.210973925243379096568082788051930123425198423068484046763181535839649628715944166508547656206673590

$g_{81}$:

-0.6591270804721472270008229642218766048162938650376074469975676463111353622379644917910511206612497946

$g_{82}$:

3.642849533154260922635760329488595445888723301481263911780293688021099480900522336187063237115090307

$g_{83}$:

-2.152093612747023427322718482636348064734352742360516656646008323679392341459513944688720683662689392

$g_{84}$:

2.698156145543009805516368225602671005762996725279378121714527188048834304066235255538342186976536315

$g_{85}$:

-1.302134502197400755385358196620807426284227923954687028400818176955088155818506805922847056566610890

$g_{86}$:

0.7166345260719169035315383971375603796337325411243794052285832676112117514575728809078560100821091090

$g_{87}$:

-1.215104683576545693698138623315868735250819547079161152214505247926780022442953298340480256959898037

$g_{88}$:

5.720470554743899462751840676746874321831380507651870511345965381060898313329871278514494993317259021

$g_{89}$:

-0.9680750549817703387755208731374516174067179237916398766071979413214745701293487163585626790764698039

$g_{90}$:

0.2842865683453974794535378501708231434170798717173266365211906641202990303773806864253211314206447484

$g_{91}$:

-2.786040212432512280783094559707904022071937624199280883230074723648044435634287586727424018338014315

$g_{92}$:

0.6557762856427716193597817942482548179689512055052806548209903916315264876908776912940420604581916545

$g_{93}$:

-3.876418032021387776313742593911123576004520372884757343393277064384243134029969022523891573825260912

$g_{94}$:

2.629405421618136304012057468949684418814178996526418965563892774361263250398486245578720920252897188

$g_{95}$:

-1.549610711913212102825487938006251348086774881841448135809034173919893130587456929323572616920181151

$g_{96}$:

0.2839850398956004115372310782922406208505218855656870355529997169694135038366453506356337752982801903

$g_{97}$:

-1.070283285176340119898411198922334910657132798886187820397090837306557805036497969882396612529460673

$g_{98}$:

5.543230604411545212916993347898352804628661852563156361292803411899484771310630093764842138517969270

$g_{99}$:

-0.9743330881854192673204178632734453200989156873015765081542748529747319487107488839304797019502931618

$g_{100}$:

1.744715568240422956522595741592227345178763394683898964808309371010365459541742295122196976944353511

$g_{101}$:

-1.315133661110433256224986569405666427255559142382576730434467574258231498658200367844750123601771313

$g_{102}$:

1.752906921727989043391493181993246889187041050786064496279784035772862926857790078769834627678528570

$g_{103}$:

-1.004030705234283069580312752217743982642645278674610633420153626288942008511580247413245395361375739

$g_{104}$:

6.049312877397079652520110007971412040386725049534562808345282436686344539911782575003454531779616339

$g_{105}$:

-0.1110016308277513824843253534266118734289609344391102138256452837215564718242494733280667078181250150

$g_{106}$:

0.9835482498560289457645749434914344420037233131380420711075543130288399829172836529736756743408786137

$g_{107}$:

-1.067437237538931291345364713754968448079517794415846712982049155340166151496293755719237230825205198

$g_{108}$:

3.040886703902520227254190761184104000063885796831912194765837735635090323162638857983871193136901144

$g_{109}$:

-3.642305292722257312964149583988018191423014640581904709310946629773423135147461476432003544435382298

$g_{110}$:

0.8576829452445604137371370040632684950338367629941977374508188985525271207717294127603672363699541403

$g_{111}$:

-2.593319239252394792079831617360287718685631676711739712586378628872201114434508392805348077904052093

$g_{112}$:

0.7935606703101061381431045109332084664969169585457400600283846375924319905763451135327677831726747768

$g_{113}$:

-0.09704718035539779781966591189595847214675288420821071353079253663337226315731500404822862941220441127

$g_{114}$:

5.498531078805297179337582649354248255246904453014254415386839084910601667278997333824881277826800855

$g_{115}$:

-2.509594611289136543367613880054578592835822767801763781379988325267684943193310625764223929107699123

$g_{116}$:

0.4940824365973205585526540398841144211064857142025402465407485368201808990524609529178109740198798233

$g_{117}$:

-0.8413569855463638685303544207971068283619650686869565758560766970345792313695329482854673095734989399

$g_{118}$:

2.541596072779034684089666168894428057269486839158109049064746125622210003619369348835212932503193147

$g_{119}$:

-1.954128911641190090833103443153376959008606285812669517750401416687842300240946544321543550080411147

$g_{120}$:

3.245157038335489078647829959653583612045914648056519928249741926877300522942243333272465555941808003

$g_{121}$:

-2.720151098294107829219517424846139523493355972009257579997647176069081741469137292398517786541394170

$g_{122}$:

0.3907447298858714268883206174230759797083760906822928533173731512425904728786535066888260944914149051

$g_{123}$:

-1.440969053614761276664470943212229925608436266507722857220265569304222374197181836873322031657386348

$g_{124}$:

0.2565147122854678643965884323932893915192362222764966000984533285807303144226619128163935972724635425

$g_{125}$:

-7.003151637297369219375170368559982123353241186097880108299233252352534456820868237710800968641754672

$g_{126}$:

-0.02762949885719993669875120344077657187547768743931447606761367233767817391098372047369064304024062313

$g_{127}$:

-1.428283405947439516990986115290923952396228122331436799174890746000487857281064027510826780105196754

$g_{128}$:

1.980396721709584060947900447903881311836683891471706642004487616751027784770517967395842615606455892

$g_{129}$:

-1.070324886694349486334308346395581150359041811740775261976350150748271687500488197524894200050893711

$g_{130}$:

2.024189659641269110593434576841245470564581265257383020526096940037376468762117679802022135290459070

$g_{131}$:

-4.467825749225196497635602798893204545015200894681228796289585023848285662305850755125105725532386269

$g_{132}$:

1.804402941187138539772983422239322744340734833700053232053767538592046324166890276329616194032403258

$g_{133}$:

-0.7763860701482502722200363690919143229922101524590044867394717310240425457533026513368392526760008650

$g_{134}$:

-0.01690039090339079218930855204964793877928124168011401157509325307844128433246264245654022777190992547

$g_{135}$:

-3.469804382462158500643888124837445546330333114732797428233719348350165491111050876039454284697877998

$g_{136}$:

3.316722321307161280895805229001378069459815965864670273703687776082154651351751548788046485054433290

$g_{137}$:

-2.279776144405432291615218857738873562527977927038156220660600771311876824003816423844725918272687644

$g_{138}$:

0.8115642624631549942882779196885160843569007991946979904897525704507087588535722954264339934207598253

$g_{139}$:

-2.433856280762851769692180853824848585903460159343634306062615264544652148468432589744710210352654782

$g_{140}$:

0.5092779498162791716294998380539613548539982084589369158750666396554761215199383342837648051403332854

$g_{141}$:

-1.101525137566525945200482079258114616586400151856445152214729163275701679020660977753931779729069142

$g_{142}$:

6.674731356276343914850437334932626676757990960278700040556307433404173024356992246601445699458519582

$g_{143}$:

-0.2209259412168413537480025244215207375848697106392529041003988356874462197446081559631949243504236026

$g_{144}$:

0.8730310682907490654484142898499069509618542867933235113735371250703318193251492673716387128804508875

$g_{145}$:

-1.382505252637439147372613470464353575807684466092711453906726819348308530876264883923635004672821421

$g_{146}$:

1.632632226159860356180831359946684763870848792558357228623193095586299361215650013366913066752181850

$g_{147}$:

-4.149058234930902949656020326947118969985537244761786649525464809192527756855996243907822368694672006

$g_{148}$:

1.242070262295007686190663408268315765942360976985428337849445324729181993923905257191999559522642263

$g_{149}$:

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0.2854653567409789096296348358539462213609048356640182975335664826916147095322882923818117675029729335

$c_{100}$:

-1.550015901707734820631200442725793145445044155629980184817964546424446693624193763193224143667720958

Definition

The Hardy Z-function, also called the Riemann-Siegel Z-function, is the real-valued function $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$ for real $t$, where $\theta(t)=\operatorname{Im}\log\Gamma(1/4+it/2)-(t/2)\log\pi$ [1]. The log-gamma branch is analytic on the right half-plane and real at $1/4$, so $\theta(0)=0$.

Parameters

$t$

— real height ($t\in\mathbb R$)

Formulas

(1)

$Z(-t)=Z(t)$ for real $t$ [1].

(2)

$Z(g_n)=(-1)^n\zeta(1/2+ig_n)$ for integers $n\geq-1$; in particular, $\zeta(1/2+ig_n)$ is real.

(3)

$Z(0)=\zeta(1/2)$, whose value is in $\zeta(1/2)$.

Comments

(4)

The Gram point $g_n$ is the unique solution in $[7,\infty)$ of $\theta(g_n)=n\pi$ for integers $n\geq-1$ [2]. The restriction excludes $t=0$ from $g_0$.

(5)

The stationary argument $c_j$ is the $j$th distinct positive real root of $Z^{\prime}(t)=0$, indexed from one in increasing order. Derivatives are taken with respect to the real height, and the origin is excluded from this sequence. All 100 represented roots are simple.

(6)

The values retain their signs. The first two stationary arguments precede the first zeta-zero ordinate; $Z(c_1)$ is a negative local maximum and $Z(c_2)$ is a negative local minimum, corresponding to Hall’s $u_{-1}$ and $u_0$ [1], page 125.

(7)

Gram’s sign rule $(-1)^n Z(g_n)>0$ fails at precisely $n=126,134,195$ within the stored Gram prefix. The phase-crossing identity does not imply a uniform sign rule.

(8)

Evenness gives the values at negative heights from those at positive heights. The origin has the value $\zeta(1/2)$.

Programs

(P1)

Python

import mpmath as mp
mp.mp.dps = 130
# Three examples; the attached generator certifies all 303 arguments.
c1 = mp.findroot(lambda t: mp.siegelz(t, derivative=1), (2, 3))
arguments = [("0", mp.mpf(0)), ("gram:-1", mp.grampoint(-1)),
             ("stationary:1", c1)]
for identity, t in arguments:
    print(identity, mp.nstr(mp.siegelz(t), 100))

References

[1]

R. R. Hall, On the stationary points of Hardy’s function Z(t), Acta Arithmetica 111 (2004), 125–140, equation (1). (doi)

Links

[2]

FLINT: Hardy Z-functions and Gram points

[3]

mpmath: Riemann-Siegel functions and Gram points

Similar tables

Gram_points_of_the_Riemann_zeta_function — stores the phase-crossing arguments used for the Gram values

Stationary_points_of_the_Hardy_Z-function — stores the stationary arguments used for the signed critical values

Hardy_Z-function_derivatives_at_Riemann_zeta_zeros — stores height derivatives of the same function at its positive zeros

Values_of_the_Riemann_zeta_function_at_rational_numbers — stores the origin value as $\zeta(1/2)$ in the real-argument zeta family

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds the origin, the Gram arguments $g_n$ for $-1\leq n\leq200$, and the stationary arguments $c_j$ for $1\leq j\leq100$; these intrinsic families cover the sign normalization, phase crossings including early failures of Gram’s sign rule, and signed critical values; the two prefix bounds are noncanonical computational cutoffs)

How they were obtained:

The attached generator evaluates FLINT Hardy Z jets [2] at 493 working bits and writes 100 significant digits. Gram and stationary arguments are recomputed as certified enclosures with ten additional requested digits; their full uncertainty is propagated into the signed values. All 303 output balls are finite and retain at least 386 relative accuracy bits.

The stationary enumeration is certified by a complete interval cover of $[0,233]$ at 96 bits. On each interval, either $Z^{\prime}$ excludes zero, or $Z^{\prime\prime}$ excludes zero and strict endpoint signs count exactly zero or one simple root. Positivity of $Z^{\prime\prime}$ near the origin excludes any positive root there. The cover proves that the selected stationary arguments are the first 100 distinct positive roots, without assuming the Riemann hypothesis. Gram arguments are certified by strict phase endpoint signs and monotonicity on $[7,\infty)$. The attached certification.md gives the arguments.

Fresh computations at higher precision are contained in every stored decimal interval. Separate log-gamma and zeta evaluations check all 303 values and the evenness relation, the Gram identity at all 202 phase crossings, and the stationary endpoint signs using digamma and zeta-prime. Independent mpmath root and value calculations at 125 and 155 decimal digits [3] agree for every entry. Known-answer controls pass, and all 606 intervals shifted by ten last-place units are rejected. These independent agreements supplement the interval proof; the reader’s mpmath example alone does not certify an error bound. The attached checks.py reproduces the checks on the stored entries.