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One-form Laplacian determinants on lens spaces
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Numbers
$p$
$q$
$u$
convention 
$D_1\text{ or }\log D_1$
2
1
0
$D_1$:
0.3305114099435124398448587491389487694887077109205723222568179420759068983668247136348795186877001104
2
1
0
$\log D_1$:
-1.107114097007906984834622741503520640194783342196192355562977724348623761176035227122416560130920492
2
1
1
$D_1$:
0.2890315556844036032809491020147637114045226306037066279580556951265058664585084330130756806920256752
2
1
1
$\log D_1$:
-1.241219407611024183461759669938962547721508838169786106674174026956074344978558891806165854852159615
3
1
0
$D_1$:
0.4841390356658170385163159111760975994344478103513900663499492641949916912099924236744594464946583421
3
1
0
$\log D_1$:
-0.7253831497454935318348702749569638017123198751619507232526080363813248591011930625985697815141032179
3
1
1
$D_1$:
0.4442023054417802101309951441012706777686433721718693445682605660041881419773684888704318571766685395
3
1
1
$\log D_1$:
-0.8114751774367188182307560682427596931019861526020138694922718574616866235267005281650063167344884445
4
1
0
$D_1$:
0.4587048054720238597448485653918054116787232628355086081529458222838007471401249265350114019712871032
4
1
0
$\log D_1$:
-0.7793484011486809247804089856992013918832890103756243364779603931636839545455832956500882517049957561
4
1
1
$D_1$:
0.5376165507909922998606351344428912972243102013591975174213087184169150305847093212806934295689383802
4
1
1
$\log D_1$:
-0.6206097038055120917308798349694812738607544190848930533370870134780371724892794459030829274260798075
4
1
2
$D_1$:
0.7205318235186226737664202607074006916956057547902954762367712362286444754295776726796109620900533377
4
1
2
$\log D_1$:
-0.3277656958592260600542137558043192483114943318205680190850173311849398066304519314723283084259247359
5
1
0
$D_1$:
0.3165092314376186727269260967147927134843618448127789856506924390145340726214471399894027116274620717
5
1
0
$\log D_1$:
-1.150402870563803733933571954163208482130649751522716942760562683643589624151416654279482394596110282
5
1
1
$D_1$:
0.4761692408830714943736328045108038761059076510901668389183788676157958057763205570875232459633856926
5
1
1
$\log D_1$:
-0.7419819398692756444825703860188310715477071039683893912113435646154041721221213158496872422607503825
5
1
2
$D_1$:
1.153748980093281373010022543331107369871239461453828302782324429437206659076896837683829417210384505
5
1
2
$\log D_1$:
0.1430166228417119273011651573791937186548858895467586314730490307848499311205325835251372320672654701
5
2
0
$D_1$:
0.7352137571521936338080829934263914416193106846074445264448924840308000731623985707663357841960067323
5
2
0
$\log D_1$:
-0.3075939959935943564435197283685963685340719458505430107455471285457040034527673386574512293708299068
5
2
1
$D_1$:
0.6003845711589696889397976947692658989293910778996495947178601142942349342718012895916942094479160952
5
2
1
$\log D_1$:
-0.5101848771563342029632156707684717048455550586288588628729011556897485256754566950677827964030625500
5
2
2
$D_1$:
0.6003845711589696889397976947692658989293910778996495947178601142942349342718012895916942094479160952
5
2
2
$\log D_1$:
-0.5101848771563342029632156707684717048455550586288588628729011556897485256754566950677827964030625500
6
1
0
$D_1$:
0.1673043098436205949368693734834939599866406296572847679612980277962335457110865203724908856121341118
6
1
0
$\log D_1$:
-1.787940910151891682297948035014696938188267884215373948399307509899795006305496007428664599768665907
6
1
1
$D_1$:
0.3160393385665472076668697849695389999658636632585990082336794596671034115543452756626989223730036210
6
1
1
$\log D_1$:
-1.151888584008711166962418714998347842098728423611604665910436750237272246091430918318130336553361152
6
1
2
$D_1$:
1.405528525203662934139946519011310714254331092891245226765049080515933113852605823320150634831141301
6
1
2
$\log D_1$:
0.3404134065719923487316626467555881489967422710095907964181648927755856225647303901531240198188727075
6
1
3
$D_1$:
2.893763084276442059556719444150964225199706530441688943547982074537365822595441044929472303996756686
6
1
3
$\log D_1$:
1.062557760406398150463077760057733136475948009053423225146699473518470147204302944830094818254562689
7
1
0
$D_1$:
0.06956174220208996916968822292224219755886406657991205793077582359787734846721837691344720915070572815
7
1
0
$\log D_1$:
-2.665540543776205610943694845925162369998362920494647367234620257768495113636995658335862974211236230
7
1
1
$D_1$:
0.1610890974903844603160502716933269580028929774125325078714031762414133997981313821605965218054816054
7
1
1
$\log D_1$:
-1.825797666505900629236796734893290748607934969025692221195049302733175169077286037391467777031677766
7
1
2
$D_1$:
1.264797423960970235970883842970249968459998748899063596389054576800197972029581812271395906543731849
7
1
2
$\log D_1$:
0.2349119701945506539425066976517533469665258464083904006100226990093201796159866543578854319953572165
7
1
3
$D_1$:
5.751664066960682785954247435405014398651800971877001100122430059911825189670681273181917688538263185
7
1
3
$\log D_1$:
1.749489215889987196617946254482876992682444492681636273083760856955753493202500152737222624650398611
7
2
0
$D_1$:
0.6487806661771602949067889337754020404501439996622041618878895780995016560376115546975079508756790947
7
2
0
$\log D_1$:
-0.4326605759819714525890969484620030056684732426333690567672933146044255784471469997517491102159290793
7
2
1
$D_1$:
0.5154495240118698170646057957540660412294471712018502931594777525983515263219380484463190362416863788
7
2
1
$\log D_1$:
-0.6627158968820751837125210790314532101914769467639208431966504223468032201622073867819560129772993754
7
2
2
$D_1$:
0.8015024290428216756008814608365098544945176398125223975466040800825139625612596344554945424097644072
7
2
2
$\log D_1$:
-0.2212672763160092222624428373729497010968126664407168651174949015236840302378725290802993779503412658
7
2
3
$D_1$:
0.9288079482196303054196031035032133939386005020320853925970429708958848455041627340088598325333815572
7
2
3
$\log D_1$:
-0.07385329112039545187867881508583717983561985566166699442078389447964901345364364372616126145593487259
8
1
0
$D_1$:
0.02310692757692589407059000374644945657240343726502575239151267788982866349446797747266050851682832586
8
1
0
$\log D_1$:
-3.767622811308920601705299524688692510058585625115259943007130552165488882810776312830514325631432751
8
1
1
$D_1$:
0.06422191970875288568700185755521259557647039680833259544734727699242457528692356656435341760280669397
8
1
1
$\log D_1$:
-2.745410698074415625947370885499521329104055433212569088017635310632829978803218042026593076222870626
8
1
2
$D_1$:
0.8488414595898474808229767498162391047059571262503949875766446876236793407856235956344568328338603977
8
1
2
$\log D_1$:
-0.1638828479296130300271068779021596241557471659102840095425086655924699033152259657361641542129623679
8
1
3
$D_1$:
8.371231399327352532610173890645670642275363342308583187037340396452671094449834161176712832290878179
8
1
3
$\log D_1$:
2.124800994268903534216491050530040055243301014127676034680548297154792806313938596123510148796790819
8
1
4
$D_1$:
19.85139754928202179695287281953187912199307173347605791592350235547721493357431129204681626214834266
8
1
4
$\log D_1$:
2.988274410160239676924890538989491118175296614739635606529170159001804928265193017180426073926436995
8
3
0
$D_1$:
0.7642258158993525003280824572271250289598192317395725342572580775763287952206852205085935621339947780
8
3
0
$\log D_1$:
-0.2688919629390952400446860470679534836897917772852126998751117228245172402455099721109924339938240443
8
3
1
$D_1$:
0.7332233976019807156387389206852894844757695892699252145529499219863215459075943019788053521466182346
8
3
1
$\log D_1$:
-0.3103048519027560458654399174847406369303772095424465266685435067390185862446397229515414637130399038
8
3
2
$D_1$:
0.8488414595898474808229767498162391047059571262503949875766446876236793407856235956344568328338603977
8
3
2
$\log D_1$:
-0.1638828479296130300271068779021596241557471659102840095425086655924699033152259657361641542129623679
8
3
3
$D_1$:
0.7332233976019807156387389206852894844757695892699252145529499219863215459075943019788053521466182346
8
3
3
$\log D_1$:
-0.3103048519027560458654399174847406369303772095424465266685435067390185862446397229515414637130399038
8
3
4
$D_1$:
0.6002215522282679053409797368830083765136649008747116210935720699372740580767670412011621313743030870
8
3
4
$\log D_1$:
-0.5104564382095856847357229386312479081934972330904116366028486703391667143000733235390958177111717117
9
1
0
$D_1$:
0.006193740463101273162015172491197412542652519866320230302696177335728210799961617860193300056726443172
9
1
0
$\log D_1$:
-5.084216099648812572548267023614175544123461411721335962921809107727442018390133124656423078871528977
9
1
1
$D_1$:
0.02029191868859971739156333178889864869452893706515459194567819255916915640614130759936893808170458978
9
1
1
$\log D_1$:
-3.897532566352560100689193159820850300754974099323776168152185563562489541038422937098410509597742046
9
1
2
$D_1$:
0.4315644553941633967199689351969423439608295447888707608455186336130557084289123706524795218447342443
9
1
2
$\log D_1$:
-0.8403384044247624373607996951934969504512412800895312542145965505246774683905575519257548084684827555
9
1
3
$D_1$:
8.841145723230918988986328627685278486916025329455547622902749874102596557587743293029717678500200000
9
1
3
$\log D_1$:
2.179416474951659520356698374328605871205570768279692619834600535673058579644470031028926648678712880
9
1
4
$D_1$:
50.72382865299173187058855790444141873914503272153569335377764434843283587091603840344652487535359516
9
1
4
$\log D_1$:
3.926395793340603719819236786771587558104229226811293552874510256625480385902279960859159001331736357
9
2
0
$D_1$:
0.3862047619245733905637296069261948106508095904497425918626480874704297324012072371495352797341105529
9
2
0
$\log D_1$:
-0.9513875788602867278359849548581586833791847959189962047007991890213660974888566984653636711808482807
9
2
1
$D_1$:
0.3036559031563432361330108644975089011597535895120964215875302354706611265122032778008725450404257465
9
2
1
$\log D_1$:
-1.191860116167736457252214089057172492790287544116562889715276004028848057219280955912525106884197825
9
2
2
$D_1$:
0.7127703947544962449934636101104816876053675456421654401239874706094264000247232684688169635790786888
9
2
2
$\log D_1$:
-0.3385959374219586163543715604934001341015884979708492263775109980447164332826173408542250543640485239
9
2
3
$D_1$:
1.119634412440510053627068726353046231738023632453142779728440456493377109600697911627046307748166709
9
2
3
$\log D_1$:
0.1130022145573965980005573399505974408334324603785227407240955763200206191938318179333969448333725314
9
2
4
$D_1$:
2.052340555285414369755834438620256618013555879017436497592454886124323561108436790858219139064924938
9
2
4
$\log D_1$:
0.7189808761529762553758295813078129337898898894853982466005151446118778669751977686017438445137579041
10
1
0
$D_1$:
0.001348790722371292240969695274774130717147447854702252819449252527349760465160702550408294933669243534
10
1
0
$\log D_1$:
-6.608546849172062448311636739354557588980127001231933612776208185088398679689801357563753726665537336
10
1
1
$D_1$:
0.005130553647320907752610628264429018188226454432193095253084617872657851789625349627155427118334211533
10
1
1
$\log D_1$:
-5.272541702168407926397787166627617538437897018359564801429448005085324358570161274646734251929552025
10
1
2
$D_1$:
0.1686645603950639952516735737345128037866900705093170762958726427987529928920601156658370851210000278
10
1
2
$\log D_1$:
-1.779843386217054550176709781683267992497518084873304781611489210100032727191156893576378426401493220
10
1
3
$D_1$:
6.840494395448862094368334376305048374774811394488721400383711736240492843401433560592956766191682823
10
1
3
$\log D_1$:
1.922860009058766477477874939062461711152403974420063413084538240884882658311689477101515658468758690
10
1
4
$D_1$:
92.81049836243684558542358062515108229495712870829806747430277465145087657304944081982932041643503509
10
1
4
$\log D_1$:
4.530559762299132281915216780608786466890189914391175410218104440469920186448039958797047009668801642
10
1
5
$D_1$:
234.6614831996825410847120783474196492087693475641346913941839859520258861981975576897980791801503292
10
1
5
$\log D_1$:
5.458143978608258714378064785191349106849477249709216670015645501444809055538384703284271332069427054
10
3
0
$D_1$:
0.5169853454845332136532034446181626040657808069371350520223735484686440494151576628655645820678818165
10
3
0
$\log D_1$:
-0.6597407501664815462120034894327468200538124732539668333006119207761675191410320464003500266977993738
10
3
1
$D_1$:
0.6714326670837223780932784985737885761257000956905919377101404348561787179892181489415352003403880419
10
3
1
$\log D_1$:
-0.3983415404459778433075608577507782498103123413933024823073097047966344651667058998872661630447822705
10
3
2
$D_1$:
0.8941843323868340325756318342751338735553146954482561979611956867629924027091435748143985284707210288
10
3
2
$\log D_1$:
-0.1118433367103563596556548130176934550352427172355563805655914508931140605087507951805166333582802795
10
3
3
$D_1$:
0.6714326670837223780932784985737885761257000956905919377101404348561787179892181489415352003403880419
10
3
3
$\log D_1$:
-0.3983415404459778433075608577507782498103123413933024823073097047966344651667058998872661630447822705
10
3
4
$D_1$:
0.8941843323868340325756318342751338735553146954482561979611956867629924027091435748143985284707210288
10
3
4
$\log D_1$:
-0.1118433367103563596556548130176934550352427172355563805655914508931140605087507951805166333582802795
10
3
5
$D_1$:
1.422117210040316736320099374171687490507491614147589576799285426286939406820684281619520839550767800
10
3
5
$\log D_1$:
0.3521467541728871897684837610641504515197405274034238225550647922304635156882647077428987973269694670
11
1
0
$D_1$:
0.0002397770108487502418882857940827516930614504112635134972886594596897832096945000120434997619489887097
11
1
0
$\log D_1$:
-8.335801187986716782533740607752524074873596572353684455754644353327453205934242211791066183917421694
11
1
1
$D_1$:
0.001045482464017385368185757873334253341259333011433627451640881328924478765628378569458176506959810006
11
1
1
$\log D_1$:
-6.863276812055586039270126782359632005517381306945914999955183907813496467283878752343460966524781436
11
1
2
$D_1$:
0.05129732116017756359411153766541262637660492703984184819105782821474444531546041189208255702816024802
11
1
2
$\log D_1$:
-2.970116747268730424239462950473574173131713481986604156651351161636691904133118807140434536734545546
11
1
3
$D_1$:
3.931118373817363585627575148985673146896227989582368328751863818131283996849566004181693980003430924
11
1
3
$\log D_1$:
1.368923958888237590315838601572289480955584002769088282484356693003917943406435147799456728198875081
11
1
4
$D_1$:
121.4132910610025128231161352610077338277784305140565567783258993396669529216133342563848935409840565
11
1
4
$\log D_1$:
4.799200354189587230185653503506760584800820442623211875704905968457381377195941536521927023914977889
11
1
5
$D_1$:
779.7731842166773793936256420016981612561527302221757329343056548070004907132515111350289759445920118
11
1
5
$\log D_1$:
6.659003087930384450126776725909176556371342539534071995176018709000266600704444921593753635612644807
11
2
0
$D_1$:
0.1646107788235801330403823907569090376399208583271379080600984261890702141370238985063238615113837931
11
2
0
$\log D_1$:
-1.804171507928291845052012230228067838805076698727562982929387207912209565467810538210132618752795283
11
2
1
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2
0
$\log D_1$:
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17
2
1
$D_1$:
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1
$\log D_1$:
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2
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5
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2
6
$D_1$:
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2
6
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2
7
$D_1$:
112.6468571841943826905713966998197072024810397922616715232204173405005774643370762531045827609958884
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2
7
$\log D_1$:
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17
2
8
$D_1$:
230.9580956076618946016436582177799493449382683907480417136028736414711609185329083456748806319399479
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2
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3
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3
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17
3
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$D_1$:
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3
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3
3
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3
3
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3
4
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3
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17
3
5
$D_1$:
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3
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3
6
$D_1$:
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3
6
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1.404096679121737366688482415446621768064808258363200348250664426167352718959114083850214743872011811
17
3
7
$D_1$:
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3
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1.809374903618932366537567579846373568809340638863464508238915293430147358024107456073044787011748716
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3
8
$D_1$:
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3
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4
0
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4
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1
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4
2
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4
5
$D_1$:
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4
6
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4
6
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4
7
$D_1$:
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4
7
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4
8
$D_1$:
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4
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5
0
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5
0
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5
1
$D_1$:
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5
2
$D_1$:
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5
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5
3
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5
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5
4
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5
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5
5
$D_1$:
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5
6
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5
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5
7
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5
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17
5
8
$D_1$:
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1
0
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1
0
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1
1
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1
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1
2
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1
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1
3
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1
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1
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1
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1
5
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1
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1
6
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1
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1
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1
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1
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1
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1
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1
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5
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5
7
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5
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5
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1
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5
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1
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1
7
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1
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21.08369315410138429170223826930712862975919294842027530143920557186079359917515595300544880187389609
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1
9
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10670678329.75574302118451658969407699647903355274502093210413058067628971495160825730698394263463356
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1
9
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23.09076547378722800194442799006434765835574187947457856589654534342666062688739636087669235314288080
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2
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2
6
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2
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2
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2
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2
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3
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3
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3
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3
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3
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3
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3
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3
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3
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3
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4
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7
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7
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7
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7
4
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7
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7
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7
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7
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1
0
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1
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1
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1
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1
4
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1
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1
6
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1
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1
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1
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1
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1
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3
6
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3
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3
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3
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30.54790654185741668980928672238644696992483721599413036705609880178618728953444138998960055479562517
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9
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9
1
$\log D_1$:
-2.636270851084203963198893583313808769218948509179782400714724002542662179285080637323367125964776012
20
9
2
$D_1$:
0.7204832689471627920252145352138467110279039671047730794503514132155938364111841998382354622276105687
20
9
2
$\log D_1$:
-0.3278330852595621426399407891200842239566706223638422858208999919856210894259548954004154065970565525
20
9
3
$D_1$:
2.615433882828786420394290710208644915442979201507779676217445272000720882176937640279125835046737697
20
9
3
$\log D_1$:
0.9614300045293832387389374695312308555762019872100317065422691204424413291558447385507578292343793452
20
9
4
$D_1$:
11.30233126288455289415627983963373282377623311265106724886442756700994534203085710211105376938217704
20
9
4
$\log D_1$:
2.425009010890006179755832658719396095855783038701268379615734860430810620466729089628763841914795976
20
9
5
$D_1$:
15.31866453708294184880044543654259974662008667628788399880973815261157946696384009061289934133311293
20
9
5
$\log D_1$:
2.729071989304129357189032392595674553424738624854608335007822750722404527769192351642135666034713527
20
9
6
$D_1$:
8.211624328090166140123110766051973192950590212202855055020202349026377829064268476352796675928462210
20
9
6
$\log D_1$:
2.105550751409126102159384121889390371034406875689907030602369580039109565981310869168283167754005666
20
9
7
$D_1$:
2.615433882828786420394290710208644915442979201507779676217445272000720882176937640279125835046737697
20
9
7
$\log D_1$:
0.9614300045293832387389374695312308555762019872100317065422691204424413291558447385507578292343793452
20
9
8
$D_1$:
0.2340992048872034856746002497391299854569812817821500285521502247057774310890606130588721300241660255
20
9
8
$\log D_1$:
-1.452010300957492407536768992563183768540847462509462495790589218114411637765201998175963019804436668
20
9
9
$D_1$:
0.07162788316934200557019054586532015978577535362809888408189026009641692583738515370648483691810640405
20
9
9
$\log D_1$:
-2.636270851084203963198893583313808769218948509179782400714724002542662179285080637323367125964776012
20
9
10
$D_1$:
0.02058482699619241903358492188178412587716002726007907061764040483866098850815992008950144005199842939
20
9
10
$\log D_1$:
-3.883201028158353979093100421342931542466966838472697508647956507291916759741163879008063830739822963
Definition
$D_1(p,q,u)$ is the zeta-regularized determinant of the full Hodge Laplacian on complex one-forms with character $\chi_u$ over the unit-radius round lens space $L(p,q)$ [5]. The determinant and its real natural logarithm are one number in two invertible conventions.
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 ($0\le u<p$)
convention
—   determinant or natural logarithm
Formulas
(1)
$\displaystyle Z_1(s)=\sum_{m=2}^\infty\left[a_{p,q,u}(m)(m^2-1)^{-s}+b_{p,q,u}(m)m^{-2s}\right]$, initially for $\Re s>3/2$, where $a_{p,q,u}(m)$ and $b_{p,q,u}(m)$ are the complex multiplicities in the exact and coexact sectors. Thus $D_1=\exp[-Z_1^\prime(0)]$ [1], equations (6)–(9).
(2)
$D_1=D_0D_\perp$ and $\log D_1=\log D_0+\log D_\perp$, where $D_0$ is the minimal scalar determinant and $D_\perp$ is the coexact one-form determinant on the same flat line [1], Section 3.
(3)
$\displaystyle T(p,q,u)=\frac{D_0^{3/2}}{D_1^{1/2}}$, where $T$ is the Ray–Singer torsion. For $u\ne0$, $T=4\sin(\pi u/p)\sin(\pi v/p)$, where $v$ is the least positive residue of $ur$ modulo $p$ and $r=q^{-1}\bmod p$. For $u=0$, $T=2\pi^2/p^2$ at unit radius [3], equation (4.3) and Proposition 5.1.
(4)
$\begin{aligned}[t]D_1(p,q,u)&=D_1(p,q,(-u)\bmod p)\\&=D_1(p,p-q,u)\\&=D_1(p,r,ur\bmod p),\end{aligned}$ where $1\le r<p$ and $qr\equiv1\pmod p$.
(5)
$\displaystyle\sum_{u=0}^{p-1}\log D_1(p,q,u)=-\log(4\pi)+\frac{3\zeta(3)}{2\pi^2}$. The sum is the full one-form logarithmic determinant on the unit three-sphere, obtained from scalar multiplicities $m^2$ and coexact multiplicities $2(m^2-1)$ [1], Section 3. Here $\zeta$ is the Riemann zeta function.
(6)
$\displaystyle\log D_1(2,1,1)=-\log2-\frac{9\zeta(3)}{2\pi^2}$, by adding the scalar and coexact formulas in [2], equation (1.1).
Comments
(7)
Let $L(p,q)=S^3/\langle g\rangle$, where $S^3\subset\mathbb C^2$ has radius one, $g(z_1,z_2)=(\zeta_pz_1,\zeta_p^qz_2)$, $\zeta_p=e^{2\pi i/p}$ and $\chi_u(g)=\zeta_p^u$. The quotient has the induced round metric. Lifted one-forms satisfy $g^*\omega=\zeta_p^u\omega$.
(8)
For the flat connection with character $\chi_u$, the nonnegative Hodge Laplacian is $\Delta_1=d_ud_u^*+d_u^*d_u$, where $d_u$ is the exterior derivative and $d_u^*$ its metric adjoint. Both exact and coexact one-forms are included, with each complex eigenspace dimension counted once. There are no harmonic one-forms for these flat lines.
(9)
Each complex eigenspace dimension is counted once. Realifying a complex line doubles multiplicities and squares the determinant, so the optional real-dimension factor in [1], Section 3, is absent. No mass term, curvature shift or unit-volume rescaling is used.
(10)
The spectral zeta function $Z_1(s)$ is continued meromorphically from $\Re s>3/2$ and is regular at zero. Its defining sum includes positive eigenvalues with their complex multiplicities. The prime on $Z_1$ denotes differentiation in $s$. The notation $\det{}^\prime\Delta_1$ agrees with $\det\Delta_1$ here because the kernel is zero.
(11)
The unoriented lens-space types are represented by the orbits of $q$ under negation and inversion modulo $p$ [4], Theorem 1.3. Conjugating the second coordinate and exchanging the coordinates realize round isometries. Exchange also changes the character according to (4). Conjugate characters have equal spectra, and residual automorphisms can give equal values in distinct rows.
(12)
The minimal scalar determinant $D_0$ and coexact one-form determinant $D_\perp$ use the same radius, quotient generator and complex characters. The nonzero exact one-form spectrum equals the scalar spectrum. Hodge star also identifies the full two-form spectrum with the full one-form spectrum [1], Section 3.
(13)
The cases $(p,q,u)=(2,1,0)$ and $(2,1,1)$ are the ordinary and nontrivially twisted full one-form determinants on real projective three-space, respectively.
(14)
The two constituent spectra have quadratic multiplicity quasipolynomials. Finite sums of the Hurwitz zeta function and its first derivative give their logarithmic determinants. The attached continuation note proves the residue-class formulas and the scalar factorization used in the computation.
Programs
(P1)
Sage
# Download generate.py from this table into the current directory.
from generate import logdet
p, q, u = 5, 2, 1
L = logdet(p, q, u, digits=100)
print(L)
print(L.exp())
References
[1]
J. S. Dowker and P. Chang, Analytic torsion on spherical factors and tessellations (2009), Section 3, equations (4), (6)–(9). (arXiv)
[2]
C. Nash and D. J. O’Connor, Determinants of Laplacians, the Ray–Singer torsion on lens spaces and the Riemann zeta function (1992), Sections 2–4, especially equation (1.1). (arXiv)
[3]
A. Kitaoka, Ray–Singer Torsion and the Rumin Laplacian on Lens Spaces, SIGMA 18 (2022), 091, 16 pp. See equation (4.3) and Proposition 5.1. (arXiv) (doi)
[4]
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, p.392. (doi)
Links
Similar tables
Scalar Laplacian determinants on lens spaces —   The nonzero exact one-form spectrum has determinant $D_0$, and $D_1=D_0D_\perp$.
Coexact one-form Laplacian determinants on lens spaces —   The coexact sector supplies $D_\perp$ in the full determinant $D_1=D_0D_\perp$.
Ray–Singer torsion of lens spaces —   The same nontrivial flat characters, with torsion $T=D_0^{3/2}/D_1^{1/2}$.
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds both conventions for 332 space-and-character pairs on 46 presentations, with every $2\le p\le20$, the smallest positive $q$ in each orbit under negation and inversion modulo $p$, and every $0\le u\le\lfloor p/2\rfloor$. This covers all unoriented types and all characters up to conjugation in the bound, including the trivial and nonfaithful characters. The cutoff in fundamental-group order is a practical, noncanonical reference extent; full type and character coverage gives breadth before extending only the homogeneous spaces $L(p,1)$)
How they were obtained:

Exact integer weight counts give the full multiplicity quasipolynomials of the exact and coexact spectra. The attached continuation note proves the finite Hurwitz-zeta formulas, including the absence of a scalar factorization anomaly. Arb evaluates the zeta values and derivatives, logarithms and exponentials at 513 working bits. All 664 values retain at least 476 bits of relative accuracy, overlap recomputations at 150 requested digits, and are stored with 100 significant digits. Each stored decimal interval contains its entire computed ball.

more

Independent 175-digit mpmath calculations use scalar and coexact trigonometric character traces and the known Ray–Singer torsion. They give two separate full one-form logarithms; the largest discrepancy from the higher-precision production computation is $4.94\times10^{-159}$. This floating comparison corroborates the values; the Arb enclosures justify the precision. Nonzero closed-form controls run first. Exact symmetry checks cover 1744 unreduced labels for each sector, character sums recover both sphere spectra, and 3320 harmonic-monomial counts independently test the scalar multiplicities. All 1328 scalar and coexact companion entries and 286 published torsions agree. The attached checks verify every read-back entry, test the Hodge and torsion identities on the stored logarithms, and check all 46 covering identities; perturbations of ten last places fail the same decimal comparison.



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