Gram points of the Riemann zeta function

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Numbers

$n$

$g_n$

-1:

9.666908056130192141261535523102232213031142486442142986903453044334597207232690144470235675455258497

17.84559954041086081682633841251909703569328743369645239211811485948168700920160952117513404084882087

23.17028270124630927899664353830153205174709832684164697083008851902296603199360939033105774834463916

27.67018221781633796093848825672068296421983963166214880461463772272318189270000981348811979235584597

31.71797995476405317955148690451646446415214307558673494959346121323725405695908241363218945211011030

35.46718429710021611605938897812867214141696593215966307332741097986362808890767200584367495141180336

38.99920996402607481744416055364232180082171095041010324698950764313946650133464529818075122072169905

42.36355039205733796940407051421783796411150932398082661508462921601797171928496299361260487198769186

45.59302898150352227397448612405307305031964061888167307056181079671638965873413052231292477217560575

48.71077662179333294037574835631652084745242468175348389305080714640077583546603507354005358290506528

51.73384281334610437069119059360365114156739972580206317626102929233757404001047197963060311297201655

10:

54.67523744685325626632662559172082511947474053852125341005321387810813652642945360229032180206332204

11:

57.54516517954725443703014058268380242426769347939691547623793247398920035264792535809838351082902328

12:

60.35181196913243699234801333601038928757464456355092536214972693987817744778155926667592268357153346

13:

63.10186798240007068491545355718099533368197402053088606328998922138334406001955807674077332636839382

14:

65.80088763805083007980428054184148342783311629130436991051307484590139578766184001364926266043460800

15:

68.45354491752273659249878842403361829240882515385097855457533623450854056640211015656152369352290470

16:

71.06381901033333997941341943478945243462435560431967012006106427704746093910373797110921676968260973

17:

73.63513225857183972710555965922963910466122811106556452330580066597632421073310613672122055410179097

18:

76.17045461108115439052946718100247307235608339796469040756127808628258322716753925954667185578431521

19:

78.67238404327764648222436173500205099209316891613695920454840697589358331465027985976392161262126747

20:

81.14320939182475716501651569711018134797073952857499000006615970526721089868840170959769344523467368

21:

83.58496009967836026249598326145298238573709521035312690535255521807057893996413212741654183461308936

22:

85.99944606665755190183920101787001776499261776784287324451353911552850270739013273305575555544825600

23:

88.38828991639700992738034541228816456256502620201707884014666342065079143947871715751418824263802112

24:

90.75295337746407476910914351645287176728833976418238412470035413240756634828307582503675723983214088

25:

93.09475904391090568955073138259576000775612440243306426112101374736833991548841042746176040567661960

26:

95.41490847053039896961960297971577885375940909382819136005710445760949068526565637384738321098542845

27:

97.71449733265807814321909085012622757919537727590007152114147689247297016117095564435231030412804413

28:

99.99452821425219901803395029423132106620826350046654639015536307908862454186580575264691956969613873

29:

102.2559214640834387988079758347082552327216176693296589755762956793431993108285845499739063004852725

30:

104.4995244664040048320825224285236989402391696897227506357006870203385738637028852763161911887663584

31:

106.7261196012284363040348017460023805954778313049988035035626726962105685996848773344221033800761186

32:

108.9364311145332391345728927865259467739829855655606982189247656319327405248305865593146058229878733

33:

111.1311310761074216263209394364370681263049646330868275992186956019220281722116246495470027742806574

34:

113.3108445694452121749471046847300957831178391521860785280165162805809993770805766025312482375513151

35:

115.4761542317570806276883857876356085113813471920177025166357100174705217480874887506309986616724486

36:

117.6276042412517472386343120788634459451235382224515207703603892974865907058610991303449609334562471

37:

119.7657038320899866429901506575742938599061983024890562927093183521852510674329059261414603887271851

38:

121.8909304039114330507327250904353267336177771680352883729686629296149301672022125394313725077685862

39:

124.0037322818898744593791453211667424072105049789894813294432880602773933868708235376695496178437189

40:

126.1045311743459311027892231532636090412161926182142662551530342389760409428152975319928342898924801

41:

128.1937243676260397311028191680059057765866957553585092422311427955335237446562147500707528761948077

42:

130.2716866919228689416281766769684355545495728777361419841095517277825706120532763419564583964489674

43:

132.3387722867140899066507563600156411180209930089099475266585136524186236696592560144704250874970950

44:

134.3953161903366318261684284077216059851416520566137579007118182420166701458913432354381091406070899

45:

136.4416357747360575414682943580196736160406721899332527144712319239082520422638149318609210769450612

46:

138.4780320435112170767683042037819282108925122679866016927195024776142172964387124476629461549991240

47:

140.5047908089133868477665136812113296127987621803374508180222038214487585051256314944735066041288635

48:

142.5221837613766280817155919745111951559018641869439297432786251472786382248004665286641289546320517

49:

144.5304694433873689769487403852988901322469388631269790801621749735719540226612581745142290015815201

50:

146.5298941379936423189327339810872935826996863043733671075630610841504921133601616044094326454431044

51:

148.5206926809650759917099373264288923015801439319941355947091173997989555885940567214322981732083437

52:

150.5030892045084616922980152449894138896396586691899707119532389023691278260110511955842153224984736

53:

152.4772978194915662910870560301438743161706714754975138467672881179858561443333517991251811301176560

54:

154.4435232423058788433218875103694731002510341467544687709788505107985768277498139626146767827724036

55:

156.4019613717873624573370696786238073057900135664190305159919243693519052409469563931716229931034393

56:

158.3527998209964861333708550971401981618065675565439464277577885754307778460755794017651471805300374

57:

160.2962184081210479706009220238590452790479369094000284454754625072284034835269551128967252508254279

58:

162.2323896102960005403395648877062827592742249539596158427323963414407389727459515729431490477437754

59:

164.1614789837239238538634539444886909228739413294799409028469498875898578265826331564186741609729747

60:

166.0836455531197544736564245007160916661948067178514215917544870106421047482172882810638315097272747

61:

167.9990421731869264671909287463163294534318307850302953053110541453730064236025561831618541981419023

62:

169.9078158645533175502707145366194324364571602133543510664105176073496196857812927519198640859429515

63:

171.8101081263493061304205798880080747120783725105660251644730395792265784782218964946325274631793294

64:

173.7060552273925516247424448299259440638892130929107196410533839517769038230983979115217346218899872

65:

175.5957884777511488881190549799689455900214437199073763063410115337929818002289254559558675441207759

66:

177.4794344822854360543687778825573576124835591581920688313448979556234447076463052832461808663617163

67:

179.3571153776162517930168255687590858508232534263825820179798460761851525135736831518368048377595428

68:

181.2289490538315139811058785544927860613270591573948728611085340990870826472151279675603227137238810

69:

183.0950493621216148786339533509210950095585939560812976460267119495014644314741778561311145740100128

70:

184.9555263094255556250811570168397901511760086412533211732205601281768198645536613859478842642384136

71:

186.8104862410724630926880614168581504370942244225692759551385452542030386194815987517290478334613251

72:

188.6600320123158297787460163770956732665217939489115431118186787197810630915435398177708129669196846

73:

190.5042631495793464884964194261916031939402578751947058447522896315443168151123858067661753043007609

74:

192.3432760021625573799637638174707018906701587421443609075157908460612239272729774463157326965731006

75:

194.1771638850908820514825583393283255662718452640684290575517150640290498743375384301009231475384967

76:

196.0060172137370523949159414582044915607892210577452293690458317173449655522152941743972832814595070

77:

197.8299236307890290807208224473833770824407106125535409230539693700619094981696280552872870680862661

78:

199.6489681260924010271097300571408550377265194012756198883047429777288270747829381552221149421844870

79:

201.4632331498526086935458807405570537837390266598134622094376087054661674222263543152679927020215394

80:

203.2727987196436074640491297812967207032723554654819922301923675344028878921769261900499411954493950

81:

205.0777425216343931041589849232488179195769562602106357188593686471779611226113087620131650980347895

82:

206.8781400064127862980016795142689156009454858250873935885395677958260998326981062642967920937530659

83:

208.6740644797566974277124474992803780235003267444820352280320321831786299872956280482242110314340503

84:

210.4655871886764816470590917952540513126137314505300707793423409959563316154855976684962466050888041

85:

212.2527774030276949263131524078272496952300110349584845665980357058972476681104750061091592715663813

86:

214.0357024929713487174114772987541314792039663616314901163799405867391092708422197479515174081148358

87:

215.8144280025384331321493021057778337593556783836654887778532044972467344728591625301879165960944123

88:

217.5890177195368564108656376909400111009515472761076724806584172254316433229761040625486019136416004

89:

219.3595337420218712962908894296643669148089963570145913386716740962459906909901710955225385386092019

90:

221.1260365415353827848062947257497480712284084736864441316471505144445907761478675465019251039219723

91:

222.8885850233051275173270708957023110857932451509023680694574755314203134632824953813032937421176456

92:

224.6472365835814668748701823527906984837120520075404419505674612731360198662583799145409671127932367

93:

226.4020471642773394380904933479647986305562268373324741271422597415327167161743600161012740313759972

94:

228.1530713050656800397311615877484070150741438843320589892195016430351402461555067360236977624902700

95:

229.9003621930782476361599084964508266813748125088085662048683399665415409802672208533350818540353962

96:

231.6439717103402363665207902663293349973424050349857452187266256003706602324907873306744271859671376

97:

233.3839504790662045535986783372004672481733325698155497026808140226330159417920142158507921996908530

98:

235.1203479049346827257297949723542419158340884563326536298800916061686057403692329219011364755679885

99:

236.8532122184512576066561271765368969077009569399183044262519278860242934029066421972568122841289729

100:

238.5825905145029233256009194384221680314690108059021767490898648798888560381044627985865199921184680

101:

240.3085287901999974871190447307722592460910134385029539814007689215819176900788035448214716956679273

102:

242.0310719810958761186444633182833123514510250711038973746384323434184144747260149324741887455344831

103:

243.7502639958693096273835275203303402720722605745077053251274284274115700577852610913659170448794669

104:

245.4661477495486869436636305500558722606491231045515127738147485245488645395567161892050196264743789

105:

247.1787651953529853120907736776344274336740334011357307978626139499056913529988682422079796475951831

106:

248.8881573552195498278560944562385297646011482350644633170776428487782162272250965514009515535537344

107:

250.5943643490846834497945205049351054161262999698650111217666852990391345606894441316030452229102772

108:

252.2974254229791307908228473554003962823694712816811046759824664477631485945827447345380144421819340

109:

253.9973789759969054984264381445679503959428424476722725118322850175727792754972789573180955676677279

110:

255.6942625861925213773570113451726103392600028324342047217438352609185220226922446620124958077924892

111:

257.3881130354585231587259643909088469243409368110730942949371472806833324680688661314891157181569675

112:

259.0789663334322571122601904106151488360121731848119699673745135549979319378866243752183886402868054

113:

260.7668577404780590608154476363904841730923909682569449966251262493997693185846703697241837990774110

114:

262.4518217897884535910150334477844194273549776487569028165334196084305132678536790198657020777925276

115:

264.1338923086455403223090722631787679592273670901602286795345320066641757504736220457533897108853573

116:

265.8131024388814790148384764212013713958400218323060390344777626045466900694543977567718259974285979

117:

267.4894846565748640414172915788331568896836536171023665321378404198424843523138113724309408311149142

118:

269.1630707910177901744813338908208962644322912013945865162989150894858376686663974653269686738900981

119:

270.8338920429865463988409307436370836159442181920814300296667195197015217614732282142088517744277570

120:

272.5019790023471239402212144365905687853182516940002758587122511288752355279674128165678889751021321

121:

274.1673616650250809506998493219582072923497316475705829304663644602485600309789135815488598841936370

122:

275.8300694493677619793390245904453976661360486021301984420286683409785356380822431027198110911138768

123:

277.4901312119254187032840819154134787377403998906729265945571059862482011923125074432178657950222071

124:

279.1475752626764131378735966810542410780567639382884394278264981250608885581129801445205704118404219

125:

280.8024293797203998905643778816311021806199922361606851704260227088694530022353748160776022410934047

126:

282.4547208234621746108397940690599354048302480007974893300380191958329646401460989208492143446756547

127:

284.1044763503077366510156230419672786152678466929348984923786492785172852974136924719623711260664120

128:

285.7517222258930404892745593455423289417806830714380538048619408130541349652699665677719811574316638

129:

287.3964842378648984093523392706859263245156788266939230908116817942589849970364925124560874305819152

130:

289.0387877082325423220462217806910415265256400196189748475636534496783236629240391865734394159318663

131:

290.6786575053074517766661142601813560839796987185655569593580180770388791333626155895357526568450174

132:

292.3161180552482047315035050624687365123058910423989095535425218299582486093325310444292318766348510

133:

293.9511933532263043580172915666812922555201061650291773198003055141350114997089940954431470517103473

134:

295.5839069742281760925879152043568411470733853001250861623107107011651138472903256660863093277077418

135:

297.2142820835078115763363138890860190867528663603426198002622633314505709020174539029134427095457563

136:

298.8423414467038574810515775789828030751551928691911251566772655021250856516858503071770629463154211

137:

300.4681074396343051262168669183090437613086909006373224261048035208612001653945874003389676480339238

138:

302.0916020577813290288756064476156311087375519357817131953981375061045480303715688743903062124185043

139:

303.7128469254782470248650477392807581087936877602021004482802295122412332594562645800513349873700033

140:

305.3318633048100294256937016773614016056357977936084995484847318370772790667533914308675780111638018

141:

306.9486721042382680274759839798869403631026236627371367564994813192924084402996571679191743803655076

142:

308.5632938869610259833807564070035077224549076277749046330794608836365706994500190921979715532691040

143:

310.1757488790175250159677343062426636547559696245517247323622208431666035455656320394777977165718965

144:

311.7860569771471857100289461510017665234499362881601212437314209093730440723211749620172155894787570

145:

313.3942377564121183147194387919393755952981487787534526719815005837165134192689259249057036297952372

146:

315.0003104775917643088295613510238759798390243350627224058972204179034233738994124741354742919700126

147:

316.6042940943580117401348281642095345241309310581442354885107395920292116681036188164292131943632678

148:

318.2062072602387489103477560513839069775000805274869706053673358327983945423334775258825977312103321

149:

319.8060683353774802838311835270948524692480618503202918077690444260650126895426402959513092173247275

150:

321.4038953930963045595910767465889286787510760898904118232837712691076148971640507048861076502339303

151:

322.9997062262692467324127111151368328106115954400372901312658211294283573716718370477837503639652581

152:

324.5935183535126428079819445053740761336107959947824703732803597447392182381935314869296968155092876

153:

326.1853490251989968095743919981814918824552334855501022469740523114643399236237657342666054558493056

154:

327.7752152293004640514118042055506841119757993477371810906210534747043262764468573172376391985397274

155:

329.3631336970678616336286466904181090967694178536713489904665276517060586716496479630994649528957903

156:

330.9491209085508660569569266148197130903327815177374356501920721787755396864885587115151837499252933

157:

332.5331930979648281232874524642546654605573476160818859167920051389679864633333508636618173206201954

158:

334.1153662589094162806684833250505778285161825626920337783956866394427003789642225342694527948094321

159:

335.6956561494440907229545311425971527382638993684051072321408097482181107597774240077091573161795513

160:

337.2740782970252113332453354578158022651519159555481726906641366477160600618271079424079094519984021

161:

338.8506480033093924654778315195111412976909229452680435722145027361016119366974978678777961247419727

162:

340.4253803488275361180798444912339448369260982474625111231569592716647363700973987348973023260907015

163:

341.9982901975338018226507890220763046944267196252426432520133717435069343812530046645589036292887309

164:

343.5693922012336061298300396955874110197235322454895282272807151050848472909065406613231214824062296

165:

345.1387008038945865283160777925521871039775754296773017932505010731996031660119046477979440231066271

166:

346.7062302458443136082263235213260698057304474356215185520032955994254600923849447420821182052990543

167:

348.2719945678583909244158434906961523863312800467759427395181301831972442747187757642835779901102500

168:

349.8360076151424439964446328093878756342950466746166959769883785634350154018332649574374823803737632

169:

351.3982830412113678854963686294402713288706023206790075893792935495412570492865148142853247222919120

170:

352.9588343116690765179345280349942565415868372667019799380735429164557591430521778269644742001579893

171:

354.5176747078918760998402864761067055486316976898927710986403263522684463155529794156802295638071963

172:

356.0748173306184693216115948052279129809876267638023650991069611350666572173883796910915182666686539

173:

357.6302751034494863356926285393900287153997862279793463839963127231300478353754259976354099208865161

174:

359.1840607762593324664454481635358533095200679499743116394100614804806540091512541905131446830847048

175:

360.7361869285230410544869655212862500541180533713670645135193181333239062794743645637194797345229699

176:

362.2866659725607225358775728334211822361863800831411843372811029216323544408451342077475975305171531

177:

363.8355101567021076080122262568262435452686942054241152629514350465839896564902879492571342356563866

178:

365.3827315683735929481883484601755519547986378862193711591664555338312054915661787568797196747206021

179:

366.9283421371101122468591768575230381770693763245261842232109811526573422431656234974761252752907578

180:

368.4723536374940731241949721192831418925491482390479630796873803501974388637373427903302421732996282

181:

370.0147776920235216533138690993495364072796489322075631020521404799662606052303393659388021957252251

182:

371.5556257739116205623260777537956152542747248122591053622279312187009155030808418767313633869529136

183:

373.0949092098194545839727232886541219799804936455611840688278095607806767098711887957655153997408284

184:

374.6326391825241067273976352856979427247497792469848793388301013237724934603906037924421483277617012

185:

376.1688267335238823297632895946120551062373639130797631882450591544562372553037532998553994210726289

186:

377.7034827655824934809475078688365552180011944460170031266246355301411445870308284115930746774756437

187:

379.2366180452139546836436922214307449653467739968332947251640276368465430027731933768803100136796689

188:

380.7682432051098813009679562251834486071487946434319458790037458740038102268565560664863534039253818

189:

382.2983687465108253468851178054248651447992598381069852081828865130052207720077585720960886510053101

190:

383.8270050415232283894298098695900430569931982698638800060958054600650634149729988350632558186621964

191:

385.3541623353835186658529762979562581268392141466998570599267179203529384802535339022647563011374860

192:

386.8798507486708288602375264224499005355021638040706879128169651868631417930079265870119848949206332

193:

388.4040802794697622800507543614586675577435995648712174149821377034161627684782766585410767545185020

194:

389.9268608054845883050274513232462403752335760453744358389666332024929041722729008356205877383652604

195:

391.4482020861062028902141930031266292503953052402631371514994883746957302236406022870068772143870757

196:

392.9681137644331465092627609148903999369658756458381952312571726343739071798761656935797511856434362

197:

394.4866053692479301520523402955247587844449185505379286522721709883027902739484996595578879481572578

198:

396.0036863169498797737730691729037113953732700814408258097598094993687811952186234979641826745939988

199:

397.5193659134456708652804527780707569860823070987228854178085838967034686164999577249605966902775274

200:

399.0336533559986875144616748105303202813621899224609573174966637403262059401100164038906321838493278

Definition

The Gram point $g_n$ is the unique real solution in $[7,\infty)$ of $\theta(g_n)=n\pi$, for integers $n\geq-1$ [1]. Here $\theta(t)=\operatorname{Im}\log\Gamma(1/4+it/2)-(t/2)\log\pi$, with the analytic log-gamma branch on the right half-plane that is real at $1/4$.

Parameters

$n$

— phase-crossing index ($n\geq-1$)

Formulas

(1)

$Z(g_n)=(-1)^n\zeta(1/2+ig_n)$, so $\zeta(1/2+ig_n)$ is real [2].

Comments

(2)

This branch gives $\theta(0)=0$ and a continuous phase. The restriction $g_n\geq7$ excludes the root $t=0$ when $n=0$ [2].

(3)

The Hardy Z-function, also called the Riemann-Siegel Z-function, is $Z(t)=e^{i\theta(t)}\zeta(1/2+it)$, which is real for real $t$ [2]. Gram points are phase landmarks used in locating the zeros of the Riemann zeta function, rather than the zero ordinates themselves.

(4)

Gram’s sign rule $(-1)^n Z(g_n)>0$ is not universal. In this prefix it fails precisely at $n=126,134,195$; in particular, $g_{126}$ is the first failure for $n\geq-1$.

Programs

[2]

Python

import mpmath as mp
mp.mp.dps = 110
for n in range(-1, 201):
    print(n, mp.nstr(mp.grampoint(n), 100))

Links

[1]

FLINT: Gram points and counting zeros

[2]

mpmath: Riemann-Siegel theta function and Gram points

[3]

DLMF 5.7.6: Partial-fraction expansion of the digamma function

[4]

DLMF 5.11.1: Stirling expansion of log-gamma

Similar tables

Zeros_of_the_Riemann_zeta_function — stores zero ordinates of the Riemann zeta function; Gram points instead mark integer multiples of its Riemann-Siegel phase

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every phase-crossing index $-1\leq n\leq200$, including the initial positive landmarks and the first failures of Gram’s sign rule; $200$ is a noncanonical cutoff for a consecutive reference prefix, chosen to cover those features without extending the sequence indefinitely)

How they were obtained:

The attached generator uses Arb log-gamma and digamma arithmetic at 493 working bits. Newton iteration locates candidates, after which opposite strict signs of $\theta(t)-n\pi$ at exact rational endpoints certify each root. The digamma series shows that $\theta'(t)$ increases for $t>0$ [3]; the certified inequality $\theta'(7)>0$ proves uniqueness on $[7,\infty)$. Stirling’s expansion gives $\theta(t)\to\infty$ [4]. The proof and endpoint construction are in certification.md. All generated enclosures retain at least 368 relative accuracy bits and are serialized to 100 significant digits.

All 202 stored decimal intervals pass fresh endpoint-sign checks, independent mpmath computations at 150 and 200 decimal digits, a phase check using Gamma itself, and the Hardy Z relation. Known-answer controls pass, and all 404 intervals shifted by ten last places are rejected. The independent agreements supplement the rigorous endpoint certificates. checks.py reproduces these checks.