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Ray–Singer torsion of lens spaces
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Numbers
$p$
$q$
$u$ 
$T(L(p,q),\chi_u)$
2
1
1:
4
3
1
1:
3
4
1
1:
2
4
1
2:
4
5
1
1:
1.381966011250105151795413165634361882279690820194237137864551377294739537181097550292792795810608863
5
1
2:
3.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
5
2
1:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
5
2
2:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
6
1
1:
1
6
1
2:
3
6
1
3:
4
7
1
1:
0.7530203962825329389499902319915203787354505382071957892689011218062926950870254308481149853468292567
7
1
2:
2.445041867912628808577805128993589518932711137529089910623974031794842464057094638149162105216145913
7
1
3:
3.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
7
2
1:
1.692021471630095869627814897002069140197260599321894121355072909988549768970069207301047119869316656
7
2
2:
1.356895867892209443894399510021300583399127186734624389483150814604022376798785292853560804220878918
7
2
3:
3.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
8
1
1:
0.5857864376269049511983112757903019214303281246230519268233202620092675215378929611496124656723584273
8
1
2:
2
8
1
3:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
8
1
4:
4
8
3
1:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
8
3
2:
2
8
3
3:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
8
3
4:
4
9
1
1:
0.4679111137620439295952146988891666521283350858392095082919094307156892222862505529435414466389030131
9
1
2:
1.652703644666139302296566746461370407999248645631861225527517243735868355721970529156966773685200852
9
1
3:
3
9
1
4:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
9
2
1:
1.347296355333860697703433253538629592000751354368138774472482756264131644278029470843033226314799148
9
2
2:
0.8793852415718167681082185546494629398724162685289292661805733255484424219917789178994917796758961349
9
2
3:
3
9
2
4:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
10
1
1:
0.3819660112501051517954131656343618822796908201942371378645513772947395371810975502927927958106088625
10
1
2:
1.381966011250105151795413165634361882279690820194237137864551377294739537181097550292792795810608863
10
1
3:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
10
1
4:
3.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
10
1
5:
4
10
3
1:
1
10
3
2:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
10
3
3:
1
10
3
4:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
10
3
5:
4
11
1
1:
0.3174929343376376622763767021612645649734150031589242027146997646366977861967154535138269546604290767
11
1
2:
1.169169973996227148941451701540753592951990179070926375147553335243120438110254020557257716168071553
11
1
3:
2.284629676546570280887585337232739337582102722251968656836165573118010979201747407803564974247345238
11
1
4:
3.309721467890570128113850144932587106367582398673856855212591209011790179898709784184341793576939660
11
1
5:
3.918985947228994779780736114132655398124909696844323910088990117990380616592573333941008561347214472
11
2
1:
1.115459702550343131946133635691985744630112543181042281688612237874890541091493387246307258079273685
11
2
2:
0.6092644793384246516668859692000682917573272981704670548763989089785904366938635497566667677702748116
11
2
3:
2.749815973232767630839284412591901805172919517773397534941436782747260178482319313383750845179142919
11
2
4:
1.967136742208932618611208635071474772608687719093044454121465808481313193005031954289738019586916161
11
2
5:
2.992228533552932465837473442771322541394167395514932652497891444375092393701994330670514838916510584
11
3
1:
1.025091791343999847226264807699847768785479676421888198376425635893779200696962376380776819329594422
11
3
2:
1.634356270682424498893150776899916060542806974592355253252824544872369637390825926137443587099869234
11
3
3:
0.8516770396585894866650749993794890279785751759120021724328535706064226519135385670434307615076424766
11
3
4:
3.601493012891357117504359411971390833151494693685399707374290353353682830395857880427181606686785395
11
3
5:
2.140551493894342979172398443391833513415592219602930480065037873768669741788455763627084077408868107
12
1
1:
0.2679491924311227064725536584941276330571947461896193719441930205480669830911999629188538132427514243
12
1
2:
1
12
1
3:
2
12
1
4:
3
12
1
5:
3.732050807568877293527446341505872366942805253810380628055806979451933016908800037081146186757248576
12
1
6:
4
12
5
1:
1
12
5
2:
1
12
5
3:
2
12
5
4:
3
12
5
5:
1
12
5
6:
4
13
1
1:
0.2290879486935802081992489559698022427890031673049301963951923825464848859325442749322354041735050528
13
1
2:
0.8638705065376883949763848817449667509330148950929636610406749968648366419176182165492715001975116698
13
1
3:
1.758926639489353893301864625094912835452637681544857190406102351966410839965879605768495303709919772
13
1
4:
2.709209774085071251939275785200036948632710864227589506843478737373520799955070896696959751107329816
13
1
5:
3.497021496342202197269261199402702767692903180351652268139262478294663414708873652187888827994965166
13
1
6:
3.941883634852104054313964552587578454499730211478007177175289052954083417520013353865149212816768523
13
2
1:
0.9502831345957173586374111601051241131800731826827323164373763854071099599891912909284644473974100436
13
2
2:
0.4448621385099018570447033531848756868068270311263549090360265746594200028111397016772603848218033572
13
2
3:
2.480121825391491043740026829230234705843707696922659310448286354827035914022526621764724346933824763
13
2
4:
1.529838690795773685102615669125110592663634514239926994010909969419925954033335330836259899536414720
13
2
5:
3.078013128314415659337579670842611703566715316385043516134614056089246775602395137315877712619256853
13
2
6:
2.633150989804513802292876317657736016759888285258688607098587481429826772791255435638617327797453496
13
3
1:
0.7878117222571309453299854142026658190601923161240627612957837409211426147538027554909290768876353500
13
3
2:
1.738094856852848303967396574307789932240265498806795077733160126328252574742994046419393524285045394
13
3
3:
0.6347825578441081867771359257751645081440117277880334646454826143183517559850739416170360960240066170
13
3
4:
2.182956995362750161012099927492665619047092529933149986769186700987672577554133748096653909106848751
13
3
5:
3.712795686158523846114715596617776211710727044173076980780096670407598531587469078932913808643263470
13
3
6:
1.845339267547382856962890903455070197699695969134625845802803740508684158037452680147688250909818146
13
5
1:
0.8950561329516654983254797433499460845196227864518935293654273551015741980482613892192238035124081026
13
5
2:
1.232673860767032802374688767387541505867019347250417670331810315580562617564942457168189461709438707
13
5
3:
1.232673860767032802374688767387541505867019347250417670331810315580562617564942457168189461709438707
13
5
4:
3.267933547648621989070012243432900524903900013046722071744070095748178528776329377255653423821460113
13
5
5:
0.8950561329516654983254797433499460845196227864518935293654273551015741980482613892192238035124081026
13
5
6:
3.267933547648621989070012243432900524903900013046722071744070095748178528776329377255653423821460113
14
1
1:
0.1980622641951617475277953609851098976681616757362856998928751536011351591441200689972770905629751694
14
1
2:
0.7530203962825329389499902319915203787354505382071957892689011218062926950870254308481149853468292567
14
1
3:
1.554958132087371191422194871006410481067288862470910089376025968205157535942905361850837894783854087
14
1
4:
2.445041867912628808577805128993589518932711137529089910623974031794842464057094638149162105216145913
14
1
5:
3.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
14
1
6:
3.801937735804838252472204639014890102331838324263714300107124846398864840855879931002722909437024831
14
1
7:
4
14
3
1:
0.8019377358048382524722046390148901023318383242637143001071248463988648408558799310027229094370248306
14
3
2:
1.356895867892209443894399510021300583399127186734624389483150814604022376798785292853560804220878918
14
3
3:
0.5549581320873711914221948710064104810672888624709100893760259682051575359429053618508378947838540873
14
3
4:
3.048917339522305313522214407023369723596387786056518510838223724592572145768854500154607924090195574
14
3
5:
2.246979603717467061050009768008479621264549461792804210731098878193707304912974569151885014653170743
14
3
6:
1.692021471630095869627814897002069140197260599321894121355072909988549768970069207301047119869316656
14
3
7:
4
15
1
1:
0.1729090847147982089957448560293656441183790812450509098780004238697702357947374566364319982134000100
15
1
2:
0.6617387872822835723474533386264390528008335620804086463650932954240667965787122057076901532798198640
15
1
3:
1.381966011250105151795413165634361882279690820194237137864551377294739537181097550292792795810608863
15
1
4:
2.209056926535306942799668309604996238161311738949186227986550953424969301386360093656360797597208852
15
1
5:
3
15
1
6:
3.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
15
1
7:
3.956295201467611275857133495739199064919475617725354215770355327281193666240190243999517050909571273
15
2
1:
0.8270909152852017910042551439706343558816209187549490901219995761302297642052625433635680017865999900
15
2
2:
0.3382612127177164276525466613735609471991664379195913536349067045759332034212877942923098467201801360
15
2
3:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
15
2
4:
1.209056926535306942799668309604996238161311738949186227986550953424969301386360093656360797597208852
15
2
5:
3
15
2
6:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
15
2
7:
2.956295201467611275857133495739199064919475617725354215770355327281193666240190243999517050909571273
15
4
1:
0.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911375
15
4
2:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
15
4
3:
1.381966011250105151795413165634361882279690820194237137864551377294739537181097550292792795810608863
15
4
4:
0.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911375
15
4
5:
3
15
4
6:
3.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
15
4
7:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
16
1
1:
0.1522409349774264877436336212064234263551667482727150277698045374389299849977952825703201302993108078
16
1
2:
0.5857864376269049511983112757903019214303281246230519268233202620092675215378929611496124656723584273
16
1
3:
1.234633135269820456543080031939202266477310875028745917132398728744907932079820615525972429315432906
16
1
4:
2
16
1
5:
2.765366864730179543456919968060797733522689124971254082867601271255092067920179384474027570684567094
16
1
6:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
16
1
7:
3.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
16
1
8:
4
16
3
1:
0.6488466976429155053447687561489003450469827504056939903090784667356404105419276543763599636430744784
16
3
2:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
16
3
3:
0.4335455026494784634546776545838784950751613763503368990535157245703375365400976785792923353730476195
16
3
4:
2
16
3
5:
3.261972627395668561058055103003274652214505127104233045406875200551802493464311756280067404028330765
16
3
6:
1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
16
3
7:
2.179580427103274592258608692270495812092361000348202156044281009245824546382286423324415105012208667
16
3
8:
4
16
7
1:
0.7653668647301795434569199680607977335226891249712540828676012712550920679201793844740275706845670943
16
7
2:
0.5857864376269049511983112757903019214303281246230519268233202620092675215378929611496124656723584273
16
7
3:
1.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
16
7
4:
2
16
7
5:
1.847759065022573512256366378793576573644833251727284972230195462561070015002204717429679869700689192
16
7
6:
3.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573
16
7
7:
0.7653668647301795434569199680607977335226891249712540828676012712550920679201793844740275706845670943
16
7
8:
4
17
1
1:
0.1350555411912883908537682163568732274748244441097661435032999762789279068607110037438505742991402980
17
1
2:
0.5219821655586817681509313802547037884801647605838045536857073786080543629084836758660449210398671673
17
1
3:
1.108523288446923465207084901241026289144644452821145013999324254296834499800541673319083767203542639
17
1
4:
1.815463281073396009520697785690987039273965432488353333616914940157724851957506088736803706996941932
17
1
5:
2.547325980144165727078155870873626863249705338853010927754858230822711489416622486337585722575963043
17
1
6:
3.205269272758512778357176309973681243237929163496828580704986988592515052744796567956083673394964981
17
1
7:
3.700434271459228304268287845898704116941320657515825694740147312835416131206850069181178662230871484
17
1
8:
3.965946199367803556563897689710397432197445750131265751994760918407815705104488434859368972258708455
17
2
1:
0.7318626990707697175574580851826398239757399063646575941379432906649866374591163976007820155790211116
17
2
2:
0.2655119279085752522956098438116933152561250926154400572546136055723995738976383656781903100278369703
17
2
3:
2.025343814585483958927224490618923074769540578269206374069150852214657126508138810471540801536095876
17
2
4:
0.9734677472556350743533166848841530616698200087113788704960242780179065929398306695752331929044023413
17
2
5:
2.857422910920880091356812788469371143052801297310120737995436664110981205303946761540285205055165815
17
2
6:
1.884970990385832294747590060207717077667355225027472361123232372677691279249343980444374955233929553
17
2
7:
3.070213731567224387503408093616808015763104719387062437201687012313587145884085564212233099095824683
17
2
8:
2.683287107199831010206244929718977454757764402913024027019279609984460689836312892090038752355097814
17
3
1:
0.6579432926143470512790204391000543799882238246438176529501287577698035633281740816184979508190019382
17
3
2:
1.153108291315062577190131975025077253691615318662814766985289082012704641790227582843592939654908441
17
3
3:
0.3869266243673933772971631638978305610053403164740384101824074023291264560477726721221943467407268692
17
3
4:
2.591910983012304839061202944657677827796676204694680680740823058538581631406308395862094895027328845
17
3
5:
2.150482918294407547043199904019410392923480317642912418377845978250090853146982346122565265261766523
17
3
6:
1.293481115514714241369766405436283250793800671904548779931207561549670489049022412870758785957074764
17
3
7:
3.830890658176515165710129473353524204722621306021499608491460942128887798243777431115518397959568157
17
3
8:
2.096745984311589313150091408732654954093284710675683566705662734295680552944254894636999906191422342
17
4
1:
0.4951649987007155259111115359250228737033914940189971140351603242429010784620535012250949888359065028
17
4
2:
1.438802691697242261871070969632600574105060886031865913755533976525876989616080813018501955372420404
17
4
3:
1.680407739882107618666929569334113811799140988378587190113614963878796945096795084992953132697801634
17
4
4:
0.4951649987007155259111115359250228737033914940189971140351603242429010784620535012250949888359065028
17
4
5:
1.680407739882107618666929569334113811799140988378587190113614963878796945096795084992953132697801634
17
4
6:
3.443964033809121788412966309455693643717280989547461198309053539799761342196004758993324051218841287
17
4
7:
3.443964033809121788412966309455693643717280989547461198309053539799761342196004758993324051218841287
17
4
8:
1.438802691697242261871070969632600574105060886031865913755533976525876989616080813018501955372420404
17
5
1:
0.7069399926264725443136128844499607501293209796672083196175906858608903521569644154177199397933992924
17
5
2:
0.7606769266092907782067213797367161889595165866344371712897739298153006523596918669032852988637434732
17
5
3:
1.418620219223637829485741818836770568947740411278254824239902687585104215687865948521783249682745411
17
5
4:
2.412270438952877336224387654516753635774880894743244784251558254543783582555911482593735148276822745
17
5
5:
0.5865411228882416970561535209863225006644796922373404603136168756887801368920579974530388461636754720
17
5
6:
3.565378730267939913414519629541830889466496213406059551236847336556488224346139065437328087931731186
17
5
7:
1.389805991685116768836478524282694203963963731008475247088072048434790200787290479219279966398023050
17
5
8:
3.178452105900546536117356465644000328461155896932021141054439934227361768298366393315133741191004317
18
1
1:
0.1206147584281832318917814453505370601275837314710707338194266744515575780082210821005082203241038651
18
1
2:
0.4679111137620439295952146988891666521283350858392095082919094307156892222862505529435414466389030131
18
1
3:
1
18
1
4:
1.652703644666139302296566746461370407999248645631861225527517243735868355721970529156966773685200852
18
1
5:
2.347296355333860697703433253538629592000751354368138774472482756264131644278029470843033226314799148
18
1
6:
3
18
1
7:
3.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
18
1
8:
3.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
18
1
9:
4
18
5
1:
0.6527036446661393022965667464613704079992486456318612255275172437358683557219705291569667736852008520
18
5
2:
0.8793852415718167681082185546494629398724162685289292661805733255484424219917789178994917796758961349
18
5
3:
1
18
5
4:
2.532088886237956070404785301110833347871664914160790491708090569284310777713749447056458553361096987
18
5
5:
0.5320888862379560704047853011108333478716649141607904917080905692843107777137494470564585533610969869
18
5
6:
3
18
5
7:
2.879385241571816768108218554649462939872416268528929266180573325548442421991778917899491779675896135
18
5
8:
1.347296355333860697703433253538629592000751354368138774472482756264131644278029470843033226314799148
18
5
9:
4
19
1
1:
0.1083655165987306419606685714301169443523762231185930412803734432112582841344149165934314384165902780
19
1
2:
0.4217189812072128015620377012018185151346273082579164432421032146501318442811986413496177833100004878
19
1
3:
0.9061036837551462505764745066076230004422290029327055198302587780968032164528245881523078089394345676
19
1
4:
1.509029025718401702155418164407258887456346881323424294871848068684723037375421185875257164703179425
19
1
5:
2.165158690944664649200687868474880455397167268679008361801055828900087974444956798922070999131155873
19
1
6:
2.803390849305938915033683319485234304513471037993440013166488731050360551615462661571591080122828298
19
1
7:
3.354563143251482149524301968991251436831039492400271713315140150716893536393378596901333540287145084
19
1
8:
3.758947502412978142781709509763682234415779778888887979005401514562833422313703407377434190941374971
19
1
9:
3.972722606805444747205018389638134221456963006405752633487330270126908132988639203256955994148291016
19
2
1:
0.6561296652262629470452697040676215679408203873555840669292077602153649370695356130468138344279764481
19
2
2:
0.2137751043924666044233088798744519870411832275168646544819287555640747106749357958795218032069160455
19
2
3:
1.845534117533080447368883804583992549374692611076847418443292082032170499017957411026076375583965659
19
2
4:
0.7977381671564156086158059351775060560898527798141124785498853348855449323184096715588763705228442897
19
2
5:
2.695025332707208273073014748055117360161094814874846971886115287839102267481047744978159641706238020
19
2
6:
1.593788811468313493581021641288801779018612510209879617204345685662745447868746608455363191810219098
19
2
7:
3.066618923050298496628543883030511221014734003473047113657071492030104916535814615104648185208856449
19
2
8:
2.381671868098726113471645618283415789378843729735523569924385516400228707334264020221973296812827810
19
2
9:
2.932844162044269347962264267789432921696412184142355270073036936066761692112179955551715756977144596
19
3
1:
0.5511722939455432344906186495060171323175684544068317001486514196665329847779159353297424601643167854
19
3
2:
1.189404452306817500323614100516370981433872223721263351514084321816805561948421797979262541155989211
19
3
3:
0.3133534646084821596013691297717015707822510851393234019617297714388735601467837247561863448934102098
19
3
4:
1.807563915860780098004330521163253766059795737726744271686274441226820158543682404334884995017135143
19
3
5:
2.852843818657831892205235003156059233973550775956182459175142736466030205860878819225126382001940403
19
3
6:
1.087310044511188900593380463205440372321719573065507851629744854034591193094222544525639381393178937
19
3
7:
2.249918476694576440626291345356423346959432897565463684133553445878110384938282221502177026238195546
19
3
8:
3.864357090206714105244349818208017277104586783287159592206956826915649848854224286663524555731700738
19
3
9:
1.897287165550792664457208812877611304071242035060734493336229952953557335162638073419283271183393731
19
4
1:
0.4843847025479334490144368054058044853076016946747890765881555634466713721716259468026900256294340798
19
4
2:
1.294361823587537212878265155077975417057124156670015718294640662365637514240041475696333915419648873
19
4
3:
1.169331757499505832171335070152899916943491968412312620320841539076547581373176541685364914025462718
19
4
4:
0.4043843591614959932574075407724307975847402864886162656902613638459398859203248104761006506542298872
19
4
5:
2.463693581087043045049600225230875334000616125082328338615482201442185095613218017381698829445111591
19
4
6:
3.246197626652751507563633397561134492478663269281678672034766707505635252258963680307902101870554806
19
4
7:
1.743439709737451847638650167273061940262539960421091918558952614249956130163758157572453215821155385
19
4
8:
1.259055007189518398624213361867257454954938265746302841970797050803284757992132210769763190191721305
19
4
9:
3.650581985814247500821040938333565290063403555770294937725028071351575138179288490784002752524784693
19
7
1:
0.6382321583612742658329954510103538491163037693144316513654329021502725771705058626495200809916724251
19
7
2:
0.6181594635539625976807164206468827846259235140054809201721901194100145965952606063556224538611459327
19
7
3:
1.400663509119671060194749592977141943103970658204831253591474625473464753241006269281825726286589147
19
7
4:
2.056793174345934007240019297044763511044791045560415320520682385688829690310541882328639560714565595
19
7
5:
0.9555566531070392277480261902784479299023087408954479658389127835124728706982407458058431108185466726
19
7
6:
3.337228521205765341219671808561863719281152470630971535763298299912701578032504766027816407631374483
19
7
7:
0.6029253419632554515789436577996358870141178783907187750415892905879198209225965977229493557637448573
19
7
8:
3.551003625598231945642980688436315706322335698147836190245227055476776288707440561907338210838290528
19
7
9:
2.448459459496335898947827462383628436388810489467566193484881372620090319940554008749025731347710516
20
1
1:
0.09788696740969285576712133324123571318860273174849955510538871113969365982961299656241437805837723665
20
1
2:
0.3819660112501051517954131656343618822796908201942371378645513772947395371810975502927927958106088625
20
1
3:
0.8244294954150537416625880907218544628046951247137080178554550384854430516752960849829919002745173281
20
1
4:
1.381966011250105151795413165634361882279690820194237137864551377294739537181097550292792795810608863
20
1
5:
2
20
1
6:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
20
1
7:
3.175570504584946258337411909278145537195304875286291982144544961514556948324703915017008099725482672
20
1
8:
3.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
20
1
9:
3.902113032590307144232878666758764286811397268251500444894611288860306340170387003437585621941622763
20
1
10:
4
20
3
1:
0.5575365158350514101328250749125074194749956954805291200090963388092964855058014653098008955360915344
20
3
2:
1
20
3
3:
0.2840790438404122960282918323931261690910880884457375827591626661550458773514845537303784177522316259
20
3
4:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
20
3
5:
2
20
3
6:
1
20
3
7:
3.520147021340201992437465501124402404531706448057263307030059911565566802989289453144792826131013901
20
3
8:
2.236067977499789696409173668731276235440618359611525724270897245410520925637804899414414408378782275
20
3
9:
1.793604493334841106541998743643783654915614055092054844279993584219817411143606364724215303914873809
20
3
10:
4
20
9
1:
0.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911375
20
9
2:
0.3819660112501051517954131656343618822796908201942371378645513772947395371810975502927927958106088625
20
9
3:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
20
9
4:
1.381966011250105151795413165634361882279690820194237137864551377294739537181097550292792795810608863
20
9
5:
2
20
9
6:
2.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
20
9
7:
1.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
20
9
8:
3.618033988749894848204586834365638117720309179805762862135448622705260462818902449707207204189391137
20
9
9:
0.6180339887498948482045868343656381177203091798057628621354486227052604628189024497072072041893911375
20
9
10:
4
Definition
This table records the positive Ray–Singer analytic torsion $T(L(p,q),\chi_u)$ of the flat complex line with nontrivial character $\chi_u$ over the three-dimensional lens space $L(p,q)$ [3].
Parameters
$p$
—   order of the fundamental group ($p\ge2$)
$q$
—   quotient-action exponent ($1\le q<p$ and $\gcd(p,q)=1$)
$u$
—   character exponent ($1\le u<p$)
Formulas
(1)
$T(L(p,q),\chi_u)=\exp\!\left(\frac12\sum_{j=0}^3(-1)^j j Z_j^\prime(0)\right)$, where $Z_j$ is the spectral zeta function of the nonnegative twisted Hodge Laplacian on complex-valued $j$-forms [1], Section 4.
(2)
$T(p,q,u)=|1-\zeta_p^u|\,|1-\zeta_p^{ur}|=4\sin(\pi u/p)\sin(\pi v/p)$, where $1\le r<p$, $qr\equiv1\pmod p$, and $v$ is the least positive residue of $ur$ modulo $p$ [1], Proposition 5.1.
(3)
$T(p,q,p-u)=T(p,q,u)$ and $T(p,p-q,u)=T(p,q,u)$.
(4)
$T(p,q,u)=T(p,r,ur\bmod p)$, where $r=q^{-1}\bmod p$.
(5)
$\prod_{u=1}^{p-1}T(p,q,u)=p^2$.
Comments
(6)
The lens space is $L(p,q)=S^3/\langle g\rangle$, where $S^3$ has unit radius, $g(z_1,z_2)=(\zeta_p z_1,\zeta_p^q z_2)$, $\zeta_p=e^{2\pi i/p}$, and $\chi_u(g)=\zeta_p^u$ [1].
(7)
The nontrivial characters are acyclic, so the torsion is independent of the metric. The metric $4g_{\mathrm{std}}$ in [1] therefore gives the same value. Complex multiplicities are used; realifying the line would square the torsion, and a reciprocal convention would invert it.
(8)
Two three-dimensional lens spaces with the same $p$ are unoriented diffeomorphic precisely when their exponents satisfy $q^\prime\equiv\pm q^{\pm1}\pmod p$ [2]. Selecting a representative fixes the generator $g$; interchanging the coordinates changes the character exponent as in (4). Nonfaithful nontrivial characters are included, and different rows may have equal torsion.
(9)
The product (2) uses roots of unity. The torsion is a positive real algebraic number.
Programs
(P1)
sage
import numberdb.sage as numberdb
from sage.rings.real_arb import RealBallField
from sage.rings.rational_field import QQ
R = RealBallField(500)
p, q, u = 7, 2, 1
v = (u * pow(q, -1, p)) % p
T = 4 * (R.pi()*R(QQ(u)/p)).sin() * (R.pi()*R(QQ(v)/p)).sin()
print(T)
References
[1]
A. Kitaoka, Ray-Singer Torsion and the Rumin Laplacian on Lens Spaces, SIGMA 18 (2022), 091, 16 pp. See pp. 2–3, Section 4, and Proposition 5.1. (arXiv) (doi)
[2]
M. Sağlam, Holomorphic curves in the symplectizations of lens spaces: An elementary approach, Münster Journal of Mathematics 15 (2022), 389–440. See Theorem 1.3. (doi)
Links
Similar tables
Roots of unity —   The cyclotomic numbers whose distances from 1 give the torsion.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds 286 values on 46 lens-space representatives, covering every $2\le p\le20$, with the smallest positive $q$ in each orbit under negation and inversion modulo $p$, and every $1\le u\le\lfloor p/2\rfloor$ for that presentation. This covers all unoriented lens-space types in the bound and all nontrivial characters up to complex conjugation, including nonfaithful ones. The cutoff is a practical initial reference extent in fundamental-group order; full type and character coverage is preferred to extending only the homogeneous spaces $L(p,1)$)
How they were obtained:

Computed with Arb sine balls at 500 bits. The smallest measured relative accuracy among the 234 irrational entries was 494 bits; each stored 100-significant-digit decimal interval contains its computed ball. Exact cyclotomic reduction of $T^2$ proves and identifies all 52 rational entries.

more

The attached checks compare every entry with a separate 175-decimal-digit log-gamma computation, whose largest relative discrepancy was $5.05\times10^{-150}$. Known values are checked first; conjugation, negation, coordinate exchange, and the full character product are also checked over the entire selection. The attached generator, argument-selection note and numerical evidence specify the computation and its finite extent.



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