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Elliptic trilogarithm L31 at CM torsion points

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Numbers
$\Delta$
$N$
$r$
$s$ 
$\mathcal L_{3,1}^{E_\Delta}(u)$
-3
1
0
0:
1.062844364839730411178751366104618430591513157432321508594484877299995816073082043115158560029465341
-3
2
0
1:
-0.7618478590774778518610533069265188731865322404603825418292681259633257187549158671605818577629401200
-3
2
1
0:
-0.01764270727615997826150510882597247487855131380692929480829776600583557164994783258789353112957944300
-3
2
1
1:
-0.01764270727615997826150510882597247487855131380692929480829776600583557164994783258789353112957944300
-3
3
0
1:
-0.4646422031370886346305280942753421826529895838697133861262851280513892431322416547888334369147891745
-3
3
1
0:
0.1732741924107815166942223045763034007380773942997277203634489597312504095620607138880190206779798460
-3
3
1
1:
0.1732741924107815166942223045763034007380773942997277203634489597312504095620607138880190206779798460
-3
3
1
2:
-0.3542814549465768037262504553682061435305043858107738361981616257666652720243606810383861866764884471
-3
4
0
1:
-0.1129339706752935073216948605816696097529354090620336030267494372037347112616305266701054107755471303
-3
4
1
0:
0.3912554182377357161210429659629512854238359149102816506259659545595063836155762707625232160409613876
-3
4
1
1:
0.3912554182377357161210429659629512854238359149102816506259659545595063836155762707625232160409613876
-3
4
1
2:
-0.3934607566472557134037311045661978447836548291361478124770031753102358300718197498360099074321588179
-3
4
1
3:
-0.3934607566472557134037311045661978447836548291361478124770031753102358300718197498360099074321588179
-3
4
2
1:
0.01770298829060877583906319721585475060461887900448578529809092145831899641726604327503267855517961526
-3
5
0
1:
0.1460799897397958397022966125438736842896743635178429774095103800409130675804663202354074353872535286
-3
5
0
2:
-0.6534610341132751189022049702982624883665774422734948997368224178168032115175210014306402925027585110
-3
5
1
0:
0.5467056092770878465838465060186146345332524882825912464279732989659916147407767695485534428340216017
-3
5
1
1:
0.5467056092770878465838465060186146345332524882825912464279732989659916147407767695485534428340216017
-3
5
1
2:
-0.2490501173185922165808290379779038869191734263877429696197260673683877776195747904095765292979472926
-3
5
1
3:
-0.5981334689942308415007898230077227280284947351072702517537635968031954032667738000912570348865221698
-3
5
1
4:
-0.2490501173185922165808290379779038869191734263877429696197260673683877776195747904095765292979472926
-3
5
2
0:
0.05273541408751655543028715139899399002363210498815788216661998572382663589592961658268846360581210820
-3
5
2
1:
0.2568392473899076887461843478428042452091662183112082143081226640383788574149081315059090690940248222
-3
5
2
2:
0.05273541408751655543028715139899399002363210498815788216661998572382663589592961658268846360581210820
-3
5
2
3:
-0.1811359206186462681389480308451595175315585268909063411510671026030761239802111111790130200949569341
-3
5
2
4:
-0.1811359206186462681389480308451595175315585268909063411510671026030761239802111111790130200949569341
-3
6
0
1:
0.3308391850221393764254013273939155332830472662734372375867115796469560100188718267016547481030985397
-3
6
1
0:
0.6550837602858902041096385994145451854134327070885720813214464708056826772874374090244350218970396665
-3
6
1
1:
0.6550837602858902041096385994145451854134327070885720813214464708056826772874374090244350218970396665
-3
6
1
2:
-0.07680855888587088209055920795806988596359188724807392917734189577110927023945828186766752591606914978
-3
6
1
3:
-0.5792552318315089925654911591901164180238321411851756998999733677833283363687125754237999802704318792
-3
6
1
4:
-0.5792552318315089925654911591901164180238321411851756998999733677833283363687125754237999802704318792
-3
6
1
5:
-0.07680855888587088209055920795806988596359188724807392917734189577110927023945828186766752591606914978
-3
6
2
1:
0.4193282089816743669758062574422943795750620638542282355033050108672174944971870745141246918395046349
-3
6
2
3:
-0.2057841727624673490648141686566563179431586116281921716940598228207921480902703690166493071350926718
-3
6
2
5:
-0.2057841727624673490648141686566563179431586116281921716940598228207921480902703690166493071350926718
-3
6
3
1:
0.008821233665338549773747371656295551853347460814423901004001133195792961165154707194985164791496670554
-3
6
3
2:
0.008821233665338549773747371656295551853347460814423901004001133195792961165154707194985164791496670554
-4
1
0
0:
1.278929236270293845234480074009576800051046400023412922560975912068827306396351530001713917880456592
comment: The square-lattice origin is $4\pi\beta(2)/9$ by (7).
-4
2
0
1:
-0.9782995703999125311392480453222065494652930080883731205728837642790052660942117361766783784123694637
-4
2
1
0:
0.6585672613323390698306280268198123494525314080825198899326397862617984394951238536762498989422553158
-4
2
1
1:
-0.6394646181351469226172400370047884000255232000117064612804879560344136531981757650008569589402282958
-4
3
0
1:
-0.5726835790076482302284041975771186576249484752157757135589285793494909702542946994089919780617438175
-4
3
1
0:
0.7608523553207964473668536771770431156542773451804935123651494332453052648653641563486815785458491711
-4
3
1
1:
-0.3782908851055282963991092006909792956926747460986728821944384074076465487269461795813367819932652527
-4
3
1
2:
-0.3782908851055282963991092006909792956926747460986728821944384074076465487269461795813367819932652527
-4
4
0
1:
-0.1127505545315502382202846138307465636194493282298416620614138113928273330236049644413726063351693702
-4
4
1
0:
0.8811821655820556860220812255019565412467527731229765647416656192423446437971343895810656949090178903
-4
4
1
1:
-0.03996653863344668266357750231279927500159520000073165383003049725215085332488598531255355993376426849
-4
4
1
2:
-0.7988612579155133022932527221494799975651863471126615785000856459596198388602439078715344575412359758
-4
4
1
3:
-0.03996653863344668266357750231279927500159520000073165383003049725215085332488598531255355993376426849
-4
4
2
1:
-0.009536891768438828172121391834529255063712297781204978010196659142048325238171502580712190966376812792
-4
5
0
1:
0.2130581648905141197033923126392575451339965648228792939162259675341410951534446040135920580162034521
-4
5
0
2:
-0.8284816449621408437531596608577371837429082543102039716570386915835343226298584845858995284732541550
-4
5
1
0:
0.9693709906695276489983549944582453868026003552233817616814620868014875721742646303540002557363370428
-4
5
1
1:
0.2267842863630482424656268758299802043380213018429449094164752424635324220109064766049400329320710012
-4
5
1
2:
-0.7107056939238724998721275675829272409502574809214493696106489840825492276047306841254342766183107721
-4
5
1
3:
-0.7107056939238724998721275675829272409502574809214493696106489840825492276047306841254342766183107721
-4
5
1
4:
0.2267842863630482424656268758299802043380213018429449094164752424635324220109064766049400329320710012
-4
5
2
0:
0.6959393394605314654544880473430760606585764958049249495738119398275299576253859366181935278690846994
-4
5
2
1:
0.1991339994157549617783355379790965209298389209120842005862586192550183050461900721247487094661281354
-4
5
2
2:
-0.5470990335890174224296479107798688207559864586096898585639497478223322041491085750046089626453834662
-4
5
2
3:
-0.5470990335890174224296479107798688207559864586096898585639497478223322041491085750046089626453834662
-4
5
2
4:
0.1991339994157549617783355379790965209298389209120842005862586192550183050461900721247487094661281354
-4
6
0
1:
0.4390639808385636530713496273709306698306871076199281860596792101012717268132957367389677675244984107
-4
6
1
0:
1.032077956826572393177840642759520343325104716388703787126216276856213577403605354496102828785187960
-4
6
1
1:
0.4253330791838153506837215705909604102000069635898523318959936208816001223797135438162591821142915911
-4
6
1
2:
-0.4979249324999263875019387645793404186858015624427139658517641072967705150638133309560286728200448067
-4
6
1
3:
-0.8858330172547193490139766599569015161535434155063567532807044529822236423781413296678480367297252128
-4
6
1
4:
-0.4979249324999263875019387645793404186858015624427139658517641072967705150638133309560286728200448067
-4
6
1
5:
0.4253330791838153506837215705909604102000069635898523318959936208816001223797135438162591821142915911
-4
6
2
1:
0.3563100171452572591175490945066144802553006584268662956015992919709053042293094218257720772007021551
-4
6
2
3:
-0.7168842060624503796890042406854011639122693097677171681193738988079688836744871420897659759648496259
-4
6
2
5:
0.3563100171452572591175490945066144802553006584268662956015992919709053042293094218257720772007021551
-4
6
3
1:
0.3193733558360428354764179796659977624315397994429660722800451916787627990401256931468216542523546065
-4
6
3
2:
-0.3289246524188703158764644588540894390435155506510624731705279672679162981627004053290454382305451540
-7
1
0
0:
1.186186969928390028209580325265579342496343112420433089149027925233902665352336309339624007874357054
-7
2
0
1:
-0.8856694348530694971119034328482927130866752555791686180877338519676378411791664605956719809470799355
-7
2
1
0:
-0.001985396296611512022640905550445896892791039368078099387018545978894578917542885704523012479343927418
-7
2
1
1:
-0.001985396296611512022640905550445896892791039368078099387018545978894578917542885704523012479343927418
-7
3
0
1:
-0.5263125462835505550907102503398739123948667564546047610664044401097025842045014435305883103728696666
-7
3
1
0:
0.1604552731610892614607644800636324965217021740068254233460602063414935211362836871971580248358601616
-7
3
1
1:
0.1604552731610892614607644800636324965217021740068254233460602063414935211362836871971580248358601616
-7
3
1
2:
-0.3217922088956902025906321876832041217580234193014607919141728282327967537802154016813384094652315694
-7
4
0
1:
-0.1126944893138313909867972553018509916669949884436918902618148957431051835409541113566127857274971648
-7
4
1
0:
0.3696907299543161278041733988702705960837117029473462906655619536461472934638336538163709962210646156
-7
4
1
1:
0.3696907299543161278041733988702705960837117029473462906655619536461472934638336538163709962210646156
-7
4
1
2:
-0.3699389044913925668070035120640763331953105828683560530889392718935091158285265145294363727809826065
-7
4
1
3:
-0.3699389044913925668070035120640763331953105828683560530889392718935091158285265145294363727809826065
-7
4
2
1:
0.001985809957197703847809326195814402531160581496295813000848164247150453393558303782153788109112172824
-7
5
0
1:
0.1844619862821057265911838516494507846227396259458562425771763898610306494983263337799902259131987752
-7
5
0
2:
-0.7535143331831088770805864740404433192650481324660403613228045142802870855273840606619212646950269252
-7
5
1
0:
0.5344276644771085391742673162856425301739870943000725079490478263868038307629903249810277533774590903
-7
5
1
1:
0.5344276644771085391742673162856425301739870943000725079490478263868038307629903249810277533774590903
-7
5
1
2:
-0.2266287411035554861106745329508258051178169805948481796390948858724189656492963839350026018365353907
-7
5
1
3:
-0.6159155100163454514681178301880338158652083020168640080281002382633066786020409466174508622221284792
-7
5
1
4:
-0.2266287411035554861106745329508258051178169805948481796390948858724189656492963839350026018365353907
-7
5
2
0:
0.05562364606674469143730830088365406571405171118993964804205469946111155252548547611822676984905329105
-7
5
2
1:
0.1943120001747742505484615620243644460825354585863183390974229570330005674634135321013016557005581107
-7
5
2
2:
0.05562364606674469143730830088365406571405171118993964804205469946111155252548547611822676984905329105
-7
5
2
3:
-0.1527795138518241755666709170048788807568534969007030239096216392068517837398975607067215606281234243
-7
5
2
4:
-0.1527795138518241755666709170048788807568534969007030239096216392068517837398975607067215606281234243
-7
6
0
1:
0.3927490135119957657333413987682515870673374553858714779575422366264049694565343379723312329504903593
-7
6
1
0:
0.6570892613806692558930346702367170501067956330318478952551200496491877047752846615736717129837508731
-7
6
1
1:
0.6570892613806692558930346702367170501067956330318478952551200496491877047752846615736717129837508731
-7
6
1
2:
-0.04856258130845487400176477371193058464719915310805679166859696347327106812031970698796623834769957917
-7
6
1
3:
-0.6086369798184983962846640274401374348994289906266307725023120569132328081511445443881669134487016049
-7
6
1
4:
-0.6086369798184983962846640274401374348994289906266307725023120569132328081511445443881669134487016049
-7
6
1
5:
-0.04856258130845487400176477371193058464719915310805679166859696347327106812031970698796623834769957917
-7
6
2
1:
0.3384693192886773999465036881862642606129158706922091772728235481211397015758009652369362837943228354
-7
6
2
3:
-0.1687937364329878057039440028443039875986432729103361902623531474920750374763528825833733181619443894
-7
6
2
5:
-0.1687937364329878057039440028443039875986432729103361902623531474920750374763528825833733181619443894
-7
6
3
1:
0.0009926981003335752923456444933269236144063059775410464211305467279359843484208723378049999145809453615
-7
6
3
2:
0.0009926981003335752923456444933269236144063059775410464211305467279359843484208723378049999145809453615
-8
1
0
0:
1.212079668994687712328781305660482948706420557452803827981862774784668203454697156252787482399367970
-8
2
0
1:
-0.9115642561268837157195283529105948423907394570361576414738986454616437028814038984537583565080302544
-8
2
1
0:
0.2843201774872452025297223423829012172819719502034628702812513217579116613138807130767384258428964572
-8
2
1
1:
-0.2818156731063772710567799687176685864210479112569080997937497573847691110234996818125706811343921801
-8
3
0
1:
-0.5392588953550983090359728910401893100633558282759040762080847510028137723182336420699483842674920128
-8
3
1
0:
0.4128943720274337745783185166477887690491460726299205783053063857770752561787311029247313061150935874
-8
3
1
1:
-0.2061687758905427799553464362839070514276552460554091017112469895615014538090703207057886792347714472
-8
3
1
2:
-0.2061687758905427799553464362839070514276552460554091017112469895615014538090703207057886792347714472
-8
4
0
1:
-0.1126934282101541994861592556908225507816288220571329196371889337931648896968443721555889036838867884
-8
4
1
0:
0.5760870475718763227038812221766384080954770901189024680629627856452890427129873953980873302420570097
-8
4
1
1:
-0.01761347956914857944104874804485428665131549445355675623710935983654806943896873011328566757089951126
-8
4
1
2:
-0.5405470253859706723876659293787757559352305963434696092778063704255500850487523062634950270116949525
-8
4
1
3:
-0.01761347956914857944104874804485428665131549445355675623710935983654806943896873011328566757089951126
-8
4
2
1:
-0.001252103805706264978781788423001804517213610072386785547048396889540573163331115151130890879616993369
-8
5
0
1:
0.1924644563297929241035686454883566939281232098962130184741244912380014391527803906212636601190968704
-8
5
0
2:
-0.7744631527639448931341322837587224818945760307752100111540828323664486460314320891390080879884651447
-8
5
1
0:
0.7038353465297327078445305322624039663425807387603707300822438987236382594026622953767047048313426517
-8
5
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1
1:
0.1419478448898285107540958893418841247423292349455460535309507232360744365872693068810736460440507748
-43
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1
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-0.05493847251789930466013438480888761787944762422988365396038815689285679961839660405225068064569896726
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-0.1740187521162268127657457571565723050023662230166369245193617324094346166459056020293169917946018377
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-0.05493847251789930466013438480888761787944762422988365396038815689285679961839660405225068064569896726
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0.002512495234968790173419392660318121699072095710146772467253701953448460610073056754691439404426258766
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1:
0.008134007434557968667964402658241726322725039754369644025010980970521394130324165069332229585524667177
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0.002512495234968790173419392660318121699072095710146772467253701953448460610073056754691439404426258766
-43
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-0.006579498952247758971104051844378907279080635494115064298484133426810419500483170981807416158176797656
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-0.006579498952247758971104051844378907279080635494115064298484133426810419500483170981807416158176797656
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0
1:
0.4006854500773219966321682474130579407657421031704242495198158761351959852604977254904372950686166927
-43
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0.2352850733736347160380202165276020339346970310788382998345871680803583093641795031865597953113661719
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0.2352850733736347160380202165276020339346970310788382998345871680803583093641795031865597953113661719
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-0.003069672825992362066924835966676507595336628584051052098100292158754282702245393165906647271198291151
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1
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-0.2322154031074924930606171680923767223414953605418445954602263659485751064122952768050839819571591123
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-0.2322154031074924930606171680923767223414953605418445954602263659485751064122952768050839819571591123
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-0.003069672825992362066924835966676507595336628584051052098100292158754282702245393165906647271198291151
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0.02449422570623546149582356045640047321292453577273325332983669751305064412752384271653889451596628170
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-0.01224710261371720675711125661774595828027246776940919476906049739341094371017528031915396083842208948
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-0.01224710261371720675711125661774595828027246776940919476906049739341094371017528031915396083842208948
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2.303865125180569630212921681147144423811158056982875413308721184653225564844312856191431251207898757e-8
-43
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2.303865125180569630212921681147144423811158056982875413308721184653225564844312856191431251207898757e-8
-67
1
0
0:
1.202056899801718656307443090003928465265526938674125550479619010125883642623660367974378046601125480
-67
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1:
-0.9015426740118200849359526231479638596313901933525725351627091630705109266603229603772864437126051072
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1
0:
-4.197344536473148471774912446588775053265108138485025472619509026537111578017485456191195014083603410e-10
-67
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1:
-4.197344536473148471774912446588775053265108138485025472619509026537111578017485456191195014083603410e-10
-67
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0
1:
-0.5342475108364374234092916471402888444656542654342284206588352068697040431185809562517180643946666983
-67
3
1
0:
0.003223312710442888464256166617429409008167147804241325026470121823126882050768177454878406251205472705
-67
3
1
1:
0.003223312710442888464256166617429409008167147804241325026470121823126882050768177454878406251205472705
-67
3
1
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-0.006446625607434422989195392762982624779803114029421140718380152388053562149026673313317879930466682691
-67
4
0
1:
-0.1126928346712119642670034158807494906056620676868475095816652751515087831483287324893798283395857952
-67
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1
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0.02420703322583633165962471468135549966305550339197901531283210943239949366641697090341494408678570664
-67
4
1
1:
0.02420703322583633165962471468135549966305550339197901531283210943239949366641697090341494408678570664
-67
4
1
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-0.02420703327830313836553907057854190524541519155779286704389492784014335649813086562863351228917564431
-67
4
1
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-0.02420703327830313836553907057854190524541519155779286704389492784014335649813086562863351228917564431
-67
4
2
1:
4.197344536500093379872540081517382935177759426863266297676949173157883624422190228755101568255030793e-10
-67
5
0
1:
0.1893679139731910494804078863934177507933707587969060550883980792499333794185244044705560361998994695
-67
5
0
2:
-0.7663552258108584919261346681651529836108345022871588559570245809055726094931987799409479463990046043
-67
5
1
0:
0.07074228339015246741139928793070954366431155857700909775155337378770491565049782577794265594282185145
-67
5
1
1:
0.07074228339015246741139928793070954366431155857700909775155337378770491565049782577794265594282185145
-67
5
1
2:
-0.02715780083730273272388872333709468673046289921458568002491259134984026781746463361280700736048002875
-67
5
1
3:
-0.08716896517285698195859150473560245293797699779879437128438491706329064920618009183895918194285030433
-67
5
1
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-0.02715780083730273272388872333709468673046289921458568002491259134984026781746463361280700736048002875
-67
5
2
0:
0.0004847368506239023739400606826866387435658048965547719640941399979079900311772387795958241647382329910
-67
5
2
1:
0.001568726972286209639755484946298333326026064969812649080118445189335056590135734508478066223074379881
-67
5
2
2:
0.0004847368506239023739400606826866387435658048965547719640941399979079900311772387795958241647382329910
-67
5
2
3:
-0.001269100336767007192955566096724651126433591381151060269396998101187300748529552858145690971909641270
-67
5
2
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-0.001269100336767007192955566096724651126433591381151060269396998101187300748529552858145690971909641270
-67
6
0
1:
0.4006856327075936139096538881777433470758773690559588419141618327438586915581972937545826642890444765
-67
6
1
0:
0.1363746205855730771459797866589924670309442965116547640459038244904089094580093038482521802115476878
-67
6
1
1:
0.1363746205855730771459797866589924670309442965116547640459038244904089094580093038482521802115476878
-67
6
1
2:
-0.0008063488026397934890924235372651604316308651014498798398725611927515325710964534190114351618656854892
-67
6
1
3:
-0.1355682718062518644150715212982456187764939829416118703966790840521110905422993253455665556668844419
-67
6
1
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-0.1355682718062518644150715212982456187764939829416118703966790840521110905422993253455665556668844419
-67
6
1
5:
-0.0008063488026397934890924235372651604316308651014498798398725611927515325710964534190114351618656854892
-67
6
2
1:
0.006447666810855404220081391646767289448114065724965615218530236676543236753962911823011280271581382997
-67
6
2
3:
-0.003223833312153379079100390323818905010575674423223887419077331805642980453786111593844429233067350372
-67
6
2
5:
-0.003223833312153379079100390323818905010575674423223887419077331805642980453786111593844429233067350372
-67
6
3
1:
2.098672268236574235887366431366816650098562367899822862042096703903692117171029420034777736052873358e-10
-67
6
3
2:
2.098672268236574235887366431366816650098562367899822862042096703903692117171029420034777736052873358e-10
-163
1
0
0:
1.202056903159589887110976371543430613241614023547903474520652125750496863069918283776266733842352884
-163
2
0
1:
-0.9015426773696913157610418311655519662321280330075130235942689798705619241836151339263345542878234369
-163
2
1
0:
-5.497860952237460104968495412423267072911481100572211553615594117894529327480469706130241384859392688e-16
-163
2
1
1:
-5.497860952237460104968495412423267072911481100572211553615594117894529327480469706130241384859392688e-16
-163
3
0
1:
-0.5342475125153730388110582879099680849116411066439544007339311851103934809270616823854360350593974238
-163
3
1
0:
0.00005774645737240572727867606333863704188104292779077921478224586270622679477935302136774667833641284853
-163
3
1
1:
0.00005774645737240572727867606333863704188104292779077921478224586270622679477935302136774667833641284853
-163
3
1
2:
-0.0001154929147450558039330071249005728350605452329175908159231402819064673602384831134180066716544613819
-163
4
0
1:
-0.1126928346712119642562254526417065112933371736345546354071830868354200257766553438345020982120179426
-163
4
1
0:
0.001391441474439837571311190834283811895265710879423327866971571998611247546170119118366776499369121944
-163
4
1
1:
0.001391441474439837571311190834283811895265710879423327866971571998611247546170119118366776499369121944
-163
4
1
2:
-0.001391441474439906294573093802535124001458366170261739260485329151255667741096592799983370005240448572
-163
4
1
3:
-0.001391441474439906294573093802535124001458366170261739260485329151255667741096592799983370005240448572
-163
4
2
1:
5.497860952237460125155143211695086155074578994643515997852537034520937102789260400129464576130357071e-16
-163
5
0
1:
0.1893679150108303247334712433213861904977820545652310545615743901987011016950427607325814000471244576
-163
5
0
2:
-0.7663552285274333825809646658628724973032893404923728141860550219671127691142756408174220553625097034
-163
5
1
0:
0.008091706293729382843834281443206698670774071740783430428868399640566275279068762732486835611534765705
-163
5
1
1:
0.008091706293729382843834281443206698670774071740783430428868399640566275279068762732486835611534765705
-163
5
1
2:
-0.003091713503856395670846041274628074217352989263901177208402594428223160487019924860465437288589338143
-163
5
1
3:
-0.009999985579746062311751716136518928402768763726026736538285944468508505705889059579149715335493150948
-163
5
1
4:
-0.003091713503856395670846041274628074217352989263901177208402594428223160487019924860465437288589338143
-163
5
2
0:
0.000003421211780551205557968057768016400182540030038855117099504993596065951794155451672209824722466497284
-163
5
2
1:
0.00001107127586220141808201506145656757596606304464357074913326662812018105787408292390268694140731999211
-163
5
2
2:
0.000003421211780551205557968057768016400182540030038855117099504993596065951794155451672209824722466497284
-163
5
2
3:
-0.000008956849711651914598975588495654215435994854155479515705165581658431846999453059953782415097047839401
-163
5
2
4:
-0.000008956849711651914598975588495654215435994854155479515705165581658431846999453059953782415097047839401
-163
6
0
1:
0.4006856343865292293221984921864655668341895876562661041845211615501737643420533448241707213044845789
-163
6
1
0:
0.02444538401721076524444693405615491847148542898105586666756791936655473544200155614494757690596643381
-163
6
1
1:
0.02444538401721076524444693405615491847148542898105586666756791936655473544200155614494757690596643381
-163
6
1
2:
-0.00001443669546075990819118397819974372764286094255852626591651325271028802091635697280969701746517528877
-163
6
1
3:
-0.02443094732175003587992770695273353567992819261219950362989943876797495990463475123106536658487939721
-163
6
1
4:
-0.02443094732175003587992770695273353567992819261219950362989943876797495990463475123106536658487939721
-163
6
1
5:
-0.00001443669546075990819118397819974372764286094255852626591651325271028802091635697280969701746517528877
-163
6
2
1:
0.0001154930769803116693321233000749170815811308098052456437753817168504265620115762806828990386711966140
-163
6
2
3:
-0.00005774653849003365997823415092536057296801856469944744875516499560944563345131967990802032983934622939
-163
6
2
5:
-0.00005774653849003365997823415092536057296801856469944744875516499560944563345131967990802032983934622939
-163
6
3
1:
2.748930476118730052484247706211633498481829636136413106731766214584824531524950317444842849058112850e-16
-163
6
3
2:
2.748930476118730052484247706211633498481829636136413106731766214584824531524950317444842849058112850e-16
Definition
The elliptic trilogarithm $\mathcal L_{3,1}^{E_\Delta}(u)=\sum_{n\in\mathbb Z}\mathcal L_3(q^n e^{2\pi i u})$ is evaluated at the torsion point $u=(r\tau_\Delta+s)/N$ of the principal marked CM torus, using the single-valued real trilogarithm $\mathcal L_3$ in (1) [1].
Parameters
$\Delta$
—   CM order discriminant ($\Delta<0$ is an imaginary quadratic order discriminant)
$N$
—   exact order of the torsion point ($N\geq1$)
$r$
—   $\tau_\Delta$ coordinate numerator ($0\leq r<N$; together with $s$, it satisfies (10))
$s$
—   real coordinate numerator ($0\leq s<N$; together with $r$, it satisfies (10))
Formulas
(1)
For $0<|z|\leq1$, $\mathcal L_3(z)=\operatorname{Re}\left(\operatorname{Li}_3(z)-\log|z|\operatorname{Li}_2(z)+\frac13\log^2|z|\operatorname{Li}_1(z)\right)$, with $\mathcal L_3(1)=\zeta(3)$. For $|z|>1$, define $\mathcal L_3(z)=\mathcal L_3(1/z)$ [1], section 1.
(2)
For $x=e^{2\pi i u}$ with $|q|<|x|\leq1$, $\mathcal L_{3,1}^{E}(u)=\mathcal L_3(x)+\sum_{n=1}^{\infty}\left(\mathcal L_3(q^n x)+\mathcal L_3(q^n/x)\right)$.
(3)
$\mathcal L_{3,1}^{E}(-u)=\mathcal L_{3,1}^{E}(u)$. For real $q$, conjugating $x$ also preserves the value: $\mathcal L_{3,1}^{E_\Delta}((r\tau_\Delta+(br-s))/N)=\mathcal L_{3,1}^{E_\Delta}((r\tau_\Delta+s)/N)$. Coordinates are interpreted modulo $N$.
(4)
For every positive integer $M$, $\sum_{r,s=0}^{M-1}\mathcal L_{3,1}^{E}((r\tau+s)/M)=M^{-2}\mathcal L_{3,1}^{E}(0)$. This is the finite-character sum of the Eisenstein-Kronecker expansion in [1], equation (2.9). The sum includes all $M$-torsion points, including those of smaller exact order.
(5)
$4\left(\mathcal L_{3,1}^{E}(1/2)+\mathcal L_{3,1}^{E}(\tau/2)+\mathcal L_{3,1}^{E}((\tau+1)/2)\right)+3\mathcal L_{3,1}^{E}(0)=0$ [1], equation (2.12).
(6)
For purely imaginary $\tau$, $8\mathcal L_{3,1}^{E}(1/4)+8\mathcal L_{3,1}^{E}(\tau/2+1/4)-\mathcal L_{3,1}^{E}(1/2)=0$ [1], equation (2.14).
(7)
For $\tau=i$, $\mathcal L_{3,1}^{E}(0)=\frac{2}{3\pi}Z_{\mathbb Z^2}(2)=\frac{4\pi}{9}\beta(2)$, where $Z_{\mathbb Z^2}(2)$ is the square-lattice Epstein zeta value and $\beta(2)=L(2,\chi_{-4})$ is the Dirichlet beta value. This follows from the square-lattice symmetry in [1], equation (2.9).
Comments
(8)
Put $b=0$ when $4\mid\Delta$ and $b=1$ otherwise, $\tau_\Delta=(-b+i\sqrt{|\Delta|})/2$, $E_\Delta=\mathbb C/(\mathbb Z+\tau_\Delta\mathbb Z)$, and $q=e^{2\pi i\tau_\Delta}$. The square root is positive. The marking is part of the convention and is not replaced by a modularly equivalent representative. In particular, $q$ is negative for odd $\Delta$.
(9)
All logarithms are natural. The functions $\operatorname{Li}_j$ are the principal classical polylogarithms, with $\operatorname{Li}_1(z)=-\log(1-z)$. On the unit circle, $\mathcal L_3(e^{i\theta})=\operatorname{Cl}_3(\theta)$, the Clausen function of order three. The limit at $z=1$ is $\zeta(3)$; zero logarithmic factors are interpreted by continuity.
(10)
The condition $\gcd(N,r,s)=1$ makes $N$ the exact order of the point. The origin is $(N,r,s)=(1,0,0)$. Among $(r,s)$ and $((-r)\bmod N,(-s)\bmod N)$, the lexicographically smaller pair is listed, with self-inverse points included once. Omitted opposite points have the same value by evenness. Signs are retained, and the origin and all points of order two are included.
(11)
The current discriminants are exactly the thirteen CM orders with rational singular modulus, including nonmaximal orders [2]. Thus $\Delta$ is an order discriminant and need not be a fundamental field discriminant. For these orders, the principal lattice represents the unique ideal class.
(12)
The Bloch elliptic dilogarithm and elliptic dilogarithm companion $J$ use the same marked CM tori and torsion coordinates at weight two. The function here has weight three and is even under point inversion. The other weight-three component $\mathcal L_{3,2}^{E}$ is a different scalar function; both components enter the K3 Mahler-measure formulas in [1], Theorem 1.2.
(13)
The full 3-torsion sum in (4) retains both $u=(\tau+1)/3$ and $u=(\tau-1)/3$. For rectangular markings these values agree by conjugation. They need not agree for rhombic markings, so the simplified relation printed in [1], equation (2.13), is not used without the rectangular restriction.
Programs
(P1)
Sage
# Download the attached generate.py into this directory first.
from generate import Table
g = Table()
print(g.value({'Delta': -4, 'N': 1, 'r': 0, 's': 0}, 100))
References
[1]
Detchat Samart, The elliptic trilogarithm and Mahler measures of K3 surfaces (2014), sections 1 and 2. (arXiv)
Links
Similar tables
Bloch elliptic dilogarithm at CM torsion points —   gives the odd weight-two elliptic polylogarithm on the same marked CM lattices
Elliptic dilogarithm companion J at CM torsion points —   gives the logarithmic weight-two component with the same torsion coordinates
Rational singular moduli —   identifies the thirteen class-number-one CM orders in the current coverage
Clausen function values —   contains the unit-circle classical trilogarithm used in the elliptic average
Classical polylogarithm values —   contains the principal classical functions from which the single-valued trilogarithm is formed
Epstein zeta values —   provides the square-lattice value in (7)
Dirichlet L-values —   contains the Dirichlet beta value in (7)
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds 494 values on the principal marked lattices of all thirteen class-number-one orders, $\Delta=-3,-4,-7,-8,-11,-12,-16,-19,-27,-28,-43,-67,-163$, at exact orders $1\leq N\leq6$, with every torsion direction represented modulo inversion. The origin and all 2-torsion points are retained. This arithmetic reference covers rectangular and rhombic CM tori, the orders $2,3,4$ in trilogarithm identities, the next prime order $5$, and the first mixed-prime order $6$. The cutoff six and lexicographic representatives are noncanonical choices independent of numeral notation; no separate application is claimed for every combination. Higher class numbers and higher torsion orders remain outside this coverage)
How they were obtained:

Every value is enclosed by Arb evaluation of the paired single-valued trilogarithm series. The CM arguments are enclosed from exact integer coordinates. For $|z|=t<1$, $|\mathcal L_3(z)|\leq\frac{t}{1-t}(1+|\log t|+\frac13\log^2 t)$ gives explicit bounds for both omitted tails by geometric sums weighted by a quadratic. Those bounds are added to the computed balls and justify the 100 significant digits recorded. The origin term uses $\zeta(3)$ directly, and unit-circle logarithmic factors are treated exactly. With 128 extra working bits, the minimum measured relative accuracy was 390 bits; between 7 and 55 terms on each side were used.

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Every entry was recomputed with 30 extra decimal working digits. A separate mpmath computation sums the polylogarithm index after analytically summing the geometric $q$-series; its unit-circle term is a finite Hurwitz-zeta sum. At 150 and 190 working decimal digits, its largest relative discrepancy between precisions was below $1.3\times10^{-134}$, and the higher-precision result differed from the Arb midpoint by less than $2.9\times10^{-148}$ relatively. A nonzero Hurwitz-zeta grouping control was checked first. Inversion and real-$q$ conjugation were recomputed at every selected input. The full torsion distribution identities for $1\leq M\leq6$ and the applicable rectangular 4-torsion identity were checked. The square-lattice origin agrees with an independent Dirichlet-beta computation. Six direct lattice sums with explicit coarse tail bounds check the Eisenstein-Kronecker normalization; they are additional agreement checks, not the certificate for the recorded digits. The attached checks.py and numerical-evidence.json give the methods and measurements.



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