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Scalar Laplacian determinants on lens spaces
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Numbers
$p$
$q$
$u$
convention 
$D_0\text{ or }\log D_0$
2
1
0
$D_0$:
2.004050458417333083781669314716021886697441120867330400882094627120002124252284040368585537045496100
2
1
0
$\log D_0$:
0.6951703617566010309682074736641203567477985469148478074167181808510466914746265536001166223260669879
2
1
1
$D_0$:
1.666051470973661273213635352396914185127351896739810504435209585112300341296897007737776731113450909
2
1
1
$\log D_0$:
0.5104564382095856847357229386312479081934972330904116366028486703391667143000733235390958177111717117
3
1
0
$D_0$:
1.325505461008831518823261157039938824381481222737020587627114539749382062909728040086399483215517525
3
1
0
$\log D_0$:
0.2817938666998530059974408085605071208126324713096027547014889779813454305010216067510623114419215443
3
1
1
$D_0$:
1.587112720819031811804875397171855546032268690394835478126449795936257617322840024997490260591328284
3
1
1
$\log D_0$:
0.4619164666331668548532448018674305720643316543478283446590389366044339876368391351940750642976585777
4
1
0
$D_0$:
0.8871238778763616839578334958304582801908635400335211830580197425774204735505890032265868217982612288
4
1
0
$\log D_0$:
-0.1197706470369173615700307696786753179157035216253031917158493880784982022748157567169250441938496835
4
1
1
$D_0$:
1.290756162477507319072408237176091628725133932332564445562263286160766657013450953929325380877996037
4
1
1
$\log D_0$:
0.2552282191047928423678614693156239540967486165452058183014243351695833571500366617695479088555858559
4
1
2
$D_0$:
2.259042404781981623739767932160837855110087943013783628017754047809276179374084350020587176121236998
4
1
2
$\log D_0$:
0.8149410087935183925382382433427956746635020685401509991325675689295448937494423103170416665199166714
5
1
0
$D_0$:
0.5821773193327278473541178687142044113744175545134161048567733934892463056407414237922460045555020076
5
1
0
$\log D_0$:
-0.5409802052609046389761452129131236858309585494050103453717673148883224171677846541512660572913441732
5
1
1
$D_0$:
0.9688399097873239379731829075523035684342278235496295681640253988645007399991169594153483219565386290
5
1
1
$\log D_0$:
-0.03165589251812738829244295988046009276415813949039756954024400360718430654363570001452407413767257389
5
1
2
$D_0$:
2.471831552625000619792258286090646631321204588863228537501172206672476049757569546834762122558379697
5
1
2
$\log D_0$:
0.9049593951316730656324807724847060681502853041955324642359110866464522180148779656597633228019640103
5
2
0
$D_0$:
0.7710177267900705369526732199293977142521552175724150881593409757176225980967142374044331004764560565
5
2
0
$\log D_0$:
-0.2600439137375015131461278043149196479654326141809523680334287965223605436015682156105890022162507147
5
2
1
$D_0$:
1.442557642046375305558050144169476027408764794874177655975928371148153712756978392683757506637771271
5
2
1
$\log D_0$:
0.3664176784259220572125145541525719782266820985465529530132489119281434873440670231874503605633723536
5
2
2
$D_0$:
1.442557642046375305558050144169476027408764794874177655975928371148153712756978392683757506637771271
5
2
2
$\log D_0$:
0.3664176784259220572125145541525719782266820985465529530132489119281434873440670231874503605633723536
6
1
0
$D_0$:
0.3691366810479391032710592497452549927271157828083765598201421510853415399468564534241265003784017939
6
1
0
$\log D_0$:
-0.9965882941822067900465612734029726821133503775218786591749841925160027811933389956667870639714909355
6
1
1
$D_0$:
0.6811567239603409314847969943542655091474027926936700808766620622307459920909814140569686760426444251
6
1
1
$\log D_0$:
-0.3839628613362370556541395716661159473662428078705348886368122500790907486971436394393767788511203840
6
1
2
$D_0$:
2.330025770855399297935167085677245315811246929314463702562445775977129146109752422686289871553483195
6
1
2
$\log D_0$:
0.8458793279694039105073843735335465194305744622183632332958511866835247363339827746334518431487789617
6
1
3
$D_0$:
3.590825645519336934641674555766932560523028396339695076407283832723195622647868415006477570658910099
6
1
3
$\log D_0$:
1.278382160882059796044002081963479802925982848831481413876473170497348211694360602417849375413412480
7
1
0
$D_0$:
0.2243258166876119189437795890096023933309408942673243130938641521409200543410411041681805121862164525
7
1
0
$\log D_0$:
-1.494655745159989171985644057123097768602174106146778442258776184214492080388859659713425744489308711
7
1
1
$D_0$:
0.4503600678438478215030320998241367829875426085142161314437768514642036959480563725804251784378020491
7
1
1
$\log D_0$:
-0.7977078654036272283411754791861608941762023041530881147812900969897597572494332484943064017937913874
7
1
2
$D_0$:
1.962747913105604946744797139587589564496826555417382001461625624057432775291193516386714265657867796
7
1
2
$\log D_0$:
0.6743454878454295084970486835896751781564274224612778908548892740806109679264093061632043365770396106
7
1
3
$D_0$:
4.364500065153336223095517523176878386794266362997281321645702955920141937579716101002483863441712676
7
1
3
$\log D_0$:
1.473503650121285663688914030305718732791509824767829167065572340611501532404803710757421157480025482
7
2
0
$D_0$:
0.4721953874637618790910371508539310172535153843535667807504032449904949622431178932406589215565931725
7
2
0
$\log D_0$:
-0.7503624225619111192007780913020446471588775468596856721030005364931355686589101068520544564908729938
7
2
1
$D_0$:
1.138498932063743193097551662540038912730124211047661142151090296035778859250089057572935513570305087
7
2
1
$\log D_0$:
0.1297106685007897285395039959510301418980765241100729030951390553248400935027470829799462322475637813
7
2
2
$D_0$:
1.138498932063743193097551662540038912730124211047661142151090296035778859250089057572935513570305087
7
2
2
$\log D_0$:
0.1297106685007897285395039959510301418980765241100729030951390553248400935027470829799462322475637813
7
2
3
$D_0$:
2.051504187859167775915524178186411755863655944690236202740697951124625070944587476565943925388498794
7
2
3
$\log D_0$:
0.7185732742624694603733462598966461722539336152123267518710055831919943002113108260357409837689282841
8
1
0
$D_0$:
0.1300211229055319886161402787669763962971820904087860135510332713596278008432529140651038091569686678
8
1
0
$\log D_0$:
-2.040058357836924333101303777952741114741469239018855399386479453736958007656139716584884838164553599
8
1
1
$D_0$:
0.2803629597290872900402477176449151988525853390736740810521779190784511592719968926545731788797088713
8
1
1
$\log D_0$:
-1.271670230517852122331785804666976459694958651791861551801182172815396908425858500696047844348988008
8
1
2
$D_0$:
1.503011112660841780690393216818342488085718948507885070794749059213575795415864125547757937190604218
8
1
2
$\log D_0$:
0.4074705043967591962691191216713978373317510342700754995662837844647724468747211551585208332599583357
8
1
3
$D_0$:
4.603875503828165133268019236231577036381105760473403702058913877097775816263019069654643779161885712
8
1
3
$\log D_0$:
1.526898449622644964699647273982600413791707268337067370102606507984980265575895162465595753204573864
8
1
4
$D_0$:
6.822921215046800380058651202784878329372150822535221867051288386192610148239578570552927084153045870
8
1
4
$\log D_0$:
1.920287710800006971531273008274065796825765717393552207670630065658459805381323959867959793970703915
8
3
0
$D_0$:
0.4173564017689547473301341802951976333534548651773229993480409692822132214768404276046127946733217038
8
3
0
$\log D_0$:
-0.8738147417136492125477659520791614392852046230755063183424731772899674601343842696783775409520173630
8
3
1
$D_0$:
1.136114502362111087171600859348722982975305201477484404230373209029483538578575785035701324565490616
8
3
1
$\log D_0$:
0.1276141095523964211839307346578119770483743082726029091507121675847916785750183308847739544277929279
8
3
2
$D_0$:
1.503011112660841780690393216818342488085718948507885070794749059213575795415864125547757937190604218
8
3
2
$\log D_0$:
0.4074705043967591962691191216713978373317510342700754995662837844647724468747211551585208332599583357
8
3
3
$D_0$:
1.136114502362111087171600859348722982975305201477484404230373209029483538578575785035701324565490616
8
3
3
$\log D_0$:
0.1276141095523964211839307346578119770483743082726029091507121675847916785750183308847739544277929279
8
3
4
$D_0$:
2.125578699922438357016261043606463201700773376957131889127668987825847344967130269924669707576145310
8
3
4
$\log D_0$:
0.7540440946767318509777351824004861213695011014502031266266237892114692578595685129614524967581676795
9
1
0
$D_0$:
0.07164966252087996410964826764538389967553072747289254773056710690864729244481083257520112465066436107
9
1
0
$\log D_0$:
-2.635966834825399596100685090221931066187735451307191594167837157317352680220103703099709794095529827
9
1
1
$D_0$:
0.1643929838209640781132776308367871457841001539394208835185208392389750229178524657155728973546433998
9
1
1
$\log D_0$:
-1.805495474758820467924779231433295479858263354409375784237316350299663465651916058119905678349768309
9
1
2
$D_0$:
1.056359832962555099614638053900271045060503283989633146008028708905424614734056113341759876882264315
9
1
2
$\log D_0$:
0.05482887815317335489305842680159148830581940451322670882073817359897854295001680617340471885029382440
9
1
3
$D_0$:
4.301141062042567802278983843820607116798195493122121329060652896229545866938970946578920353374437144
9
1
3
$\log D_0$:
1.458880350762626301049062949391219093500183961308397174434663067649349055360562654925386052768725686
9
1
4
$D_0$:
9.139293302165061080959996424421946734282817879855345303682621194150166344284528738089802488037535603
9
1
4
$\log D_0$:
2.212583063238813967884965606499134563616775604243977420075617113305118910338738387140576023797133062
9
2
0
$D_0$:
0.2841203599679476952152629413841727587225927005734151283097815430098692855126967500288225265638656516
9
2
0
$\log D_0$:
-1.258357327895890981196591067303258779272976579373078341427500517748660706586344894369356658198636262
9
2
1
$D_0$:
0.8199159458977415185459679927759766885204012033047286211591386725544804928571814362296159843448709627
9
2
1
$\log D_0$:
-0.1985534489805893389589854466445941652757155777028063761063290686840534375403253247539646115267159501
9
2
2
$D_0$:
0.8199159458977415185459679927759766885204012033047286211591386725544804928571814362296159843448709627
9
2
2
$\log D_0$:
-0.1985534489805893389589854466445941652757155777028063761063290686840534375403253247539646115267159501
9
2
3
$D_0$:
2.159929532449371422049030189092652228916498256108111219088981565891308916426295575843007676894495683
9
2
3
$\log D_0$:
0.7700755972978719935970159379318829500428045253413405480644947478650030685436832505602094848202789030
9
2
4
$D_0$:
2.360853873947353110037827550691010213020132739100515662602497115925727832185658685803865231817503324
9
2
4
$\log D_0$:
0.8590233645943455327712156951566189026157628097534410968716970739725408627174897847020042873510904779
10
1
0
$D_0$:
0.03745714602218165016070434094039959087783531192049491687228424650209844187316204643368332702896459926
10
1
0
$\log D_0$:
-3.284557772210251289658476303254475478881451145121756240871222494694450264973505842720507603976378108
10
1
1
$D_0$:
0.09079694244374066485127146787323862480554705842490657223478677635182107127216719831704265085677499835
10
1
1
$\log D_0$:
-2.399129667468940572129607606775030410326211451967402293357840226815326609667682790764923989473481975
10
1
2
$D_0$:
0.6854979928674255705124103823120413113742963507001473064927265272749569326868703141519922237977409048
10
1
2
$\log D_0$:
-0.3776097079673870235238227584352723997474284664587026996736258854353938248999808925900878021845868531
10
1
3
$D_0$:
3.605891743439502178779916807275734363642615989638853001588051817925045281543176205335119003258037745
10
1
3
$\log D_0$:
1.282569103099060089156303530919978467897713770654235163909536972081846042914858858249851124986550863
10
1
4
$D_0$:
10.67040236941495778941990313653062275469717843659293580830527922545623953313248089350225648052765555
10
1
4
$\log D_0$:
2.367473774950813183837164646894570317562053312477004723817596223208142303124047090750399915335809401
10
1
5
$D_0$:
15.54249004950803700980588055472074954218840301950256890035104496538971555879576382853945730566778267
10
1
5
$\log D_0$:
2.743577566949346650682331090341351793050492595716745895499455179806127847805721188569241546685033935
10
3
0
$D_0$:
0.2720899943371634746483990058465410385907578619994233880801377543667830532675564537082672009338455649
10
3
0
$\log D_0$:
-1.301622405875057655625265219947205222572679635795767314379357073257039878123916072332706370653798787
10
3
1
$D_0$:
0.8756572659824908172641212430261327474659424572024179169406959134891193692216717418763570443198692463
10
3
1
$\log D_0$:
-0.1327805134819926144358536192502594166034374471311008274357699015988781550555686332957553876815940902
10
3
2
$D_0$:
1.647399842480402521366273219130417532660490129772279551536779529353138784613959023026987411493972369
10
3
2
$\log D_0$:
0.4991981919079146716483681734028313948301195456776537804490188135270216423996356564832057482449664438
10
3
3
$D_0$:
0.8756572659824908172641212430261327474659424572024179169406959134891193692216717418763570443198692463
10
3
3
$\log D_0$:
-0.1327805134819926144358536192502594166034374471311008274357699015988781550555686332957553876815940902
10
3
4
$D_0$:
1.647399842480402521366273219130417532660490129772279551536779529353138784613959023026987411493972369
10
3
4
$\log D_0$:
0.4991981919079146716483681734028313948301195456776537804490188135270216423996356564832057482449664438
10
3
5
$D_0$:
2.833686437710954455568857628558784206464915632484906088769251438685564597350228593570456742784544177
10
3
5
$\log D_0$:
1.041578492137556142479137415632285574607247021614814946345928276734679334522347856722117368437548072
11
1
0
$D_0$:
0.01854807148843327537067205187302330906782045342970095439475480039480227777591403524272331537047317354
11
1
0
$\log D_0$:
-3.987389458220902547791113757094817763806748155907198787444354294991469792702831771248355093611054885
11
1
1
$D_0$:
0.04723475068368642258251980516710159558480505283318029101800464438785787502037051985331448577521828002
11
1
1
$\log D_0$:
-3.052625413986328940303323072345067319056790646034765889990019517410531539853059302196704774289911668
11
1
2
$D_0$:
0.4123664566765292541927195231331315351748429440014598602676545053558755843658731147878172300582909768
11
1
2
$\log D_0$:
-0.8858428670131903913307119437583681505517967840745590930728790334129003079881554649855160536574190527
11
1
3
$D_0$:
2.737679368309069832016134544659288002652258541405642948587620973236723537894736111777761236600427990
11
1
3
$\log D_0$:
1.007110615666082678029691152536757786910075823922713092022747732104338356503346160587018217280104482
11
1
4
$D_0$:
10.99722325438900194505901098403234210304278849842932530684628245215629341289281920775432401889732003
11
1
4
$\log D_0$:
2.397642809513015412585258305450877775610908532831677903701485079700827449627153161319509736786094808
11
1
5
$D_0$:
22.87908066351652440976154835292473249412367014476864748044303677292196657167817105327271302322000001
11
1
5
$\log D_0$:
3.130222984913965872766607642810892921461625041311163103070626312109107640949481269469476640705278223
11
2
0
$D_0$:
0.1636247529074526029554079878210079005496627339739969188490471867829785570699152108916125271820433276
11
2
0
$\log D_0$:
-1.810179564868094235297204297919999018450574864698491629835935246519721912547354546721377238556179414
11
2
1
$D_0$:
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1
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1
5
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1
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0.6624725364059951145900243013385442384818992117344885219172205759488281247970220354087995307314108456
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6.224196118482148043830938444916927573973903490559018004141483051375347292832793298576555632285087232
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1.828444296418565164926805102381246008563398409298034461779520280332317657833683255848249759924312301
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1
5
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33.61191778745496502230935815124547921637699785764296812602687735937018091880210496106687510643482441
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3.514880700134590783280154546361652484253432423869518740256282432499204431800755319547735315431052070
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1
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4.575460288572236123252621812810249690507720051960321469193835824591346411392258580570952903938882116
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1
7
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139.3393937169154682782059587880436669544530532582686615455834663626680284338841656811500771355924815
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1
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4.936912638486689823838790574051818660598254266638603913468431527944602438839705432755801003202988216
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0
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0.5521684354869495880084637878410614875099943345723430811079779649602658223280242132707953543022966569
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3
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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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4
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4
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4
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4
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5
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5
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5
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5
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5
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5
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5
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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
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1
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2.473378837113098057467780899278445322927885348514304047253922554442451958089959476930828131163895540
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1
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115.5714920887624788137448350604330399436964446583975283178644214907820761205692830994153813722524818
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1
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4.749889317596769448840005947389146609611299065283123602032263714583255766006703778933817247999507111
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1
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1
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1
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1
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7.504871873557514385075715931019565940833355221485362157232991579615063228745128369620391785339298052
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1
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2592.557436039205119961488069898730065516456415598418982094135015746419911660516157583662220463648876
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1
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7.860400094550346091625888077959334835148213056780934269071674464365161632986642850434711765874975330
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5
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5
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5
5
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5
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5
6
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5
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5
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5
8
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1.528322108388424923721599307167817805072009401899214759189453054176036449865889925894239993281171247
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5
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0.4241704724474309436521018792718108995916413284304066883215778965585301827674381940542006231362845374
18
5
9
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5
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1.024433930553421492181061183784904785827513933391019847132785154276353602758684340808304574040521370
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1
0
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1
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1
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1
5
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1.968089629449620375354693154358408539180664306184341303064725932340909481970629156280092296813563583
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1
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91.47829764265580177551639574555823894886811546764371414922585816476337768626966246905866056343572036
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4.516101759915576950267903701388526048497014987141107047291572557527939197844494122988063682002656738
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1
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682.3716165213751088402506637232252621620815284271813265851705031557695162079637737526666722508745523
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1
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6.525574401703660873794308896497185273092145745410607388057493677626450243389877797224480313745228136
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1
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2726.181116571410304625779083592948964884159156235094190246113423442614770023911098135956053226721909
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7.910657050464455438501171607635165939944292814736406053931655137094718138800486152950481867627551701
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1
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5522.336126906217429458124844243065959414971635656466480957364869797104405704101497228577847896373921
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1
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2
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2
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2
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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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4
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4
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4
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4
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4
9
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7
0
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7
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7
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7
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7
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7
7
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7
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7
8
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7
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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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1
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1
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1
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1
10
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1
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3
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3
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3
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3
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0.5207892460687780712395687078161427873036235108074074731729641383673396672611739283610586842187740362
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3
6
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1.505805883564374042904415017663908377864353822502944846536840913969845267295582407597181230278751616
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0.4093282256943973746224083551164185598257478543445075647311920469593613484425269384718432112028783768
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3
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4.614715587613623175418184122106645096666485969822190183975578470908048096184137287933720964177389119
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3
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1.529250238236376803265316516305958441133292439959857946429717494225652822299908725255628492948596272
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3
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3
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1.533505441392327639519236558559518104409924694490127774700881273088151896781774041247390824410258492
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3
9
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3
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1.529250238236376803265316516305958441133292439959857946429717494225652822299908725255628492948596272
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3
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3
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2.064899056373549579035158303423331974171616657476083858173908394133082705628665202901725396444744014
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9
0
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9
1
$\log D_0$:
-1.199564833734470286064803803387515205163105725983701146678920113407663304833841395382461994736740987
20
9
2
$D_0$:
0.4719446997379575442957919984925836262134063185759850092684506728523550063678344773250807190333698798
20
9
2
$\log D_0$:
-0.7508934618326586442103254809391859721658026533021614548216575557820921866196139976828177076959834842
20
9
3
$D_0$:
1.898918572092943162486294956259656966904235634090278752847492195724831926327120180194776381614440213
20
9
3
$\log D_0$:
0.6412845515495300445781517654599892339488568853271175819547684860409230214574294291249255624932754317
20
9
4
$D_0$:
2.784331358209567721798373174277338558026532305441995299448631167262275624820069284236109751801326086
20
9
4
$\log D_0$:
1.024007757734966553120358055032282296370338574732821687402115471408220624319314435144959620587509546
20
9
5
$D_0$:
3.942396485579302690869547199673912475310833868032121152390038830967036571781326787043053725998374986
20
9
5
$\log D_0$:
1.371788783474673325341165545170675896525246297858372947749727589903063923902860594284620773342516967
20
9
6
$D_0$:
3.832303342040595840887758228054655635890808369250450846200576679501501939221596791252576299569089661
20
9
6
$\log D_0$:
1.343466017215846630716806591862288021191714737744183036415480751799921678804732655605440294748299855
20
9
7
$D_0$:
1.898918572092943162486294956259656966904235634090278752847492195724831926327120180194776381614440213
20
9
7
$\log D_0$:
0.6412845515495300445781517654599892339488568853271175819547684860409230214574294291249255624932754317
20
9
8
$D_0$:
1.452496432840630062538198474977482937821183061031982855939563201869096157710855037208502094475294602
20
9
8
$\log D_0$:
0.3732837538652716206865027225039135724183741868434587551480316703466983617196331050927299055113966310
20
9
9
$D_0$:
0.3013253099952618730422097931736475879260961225507732586547739942468632681388229965293590563120520303
20
9
9
$\log D_0$:
-1.199564833734470286064803803387515205163105725983701146678920113407663304833841395382461994736740987
20
9
10
$D_0$:
0.6905933647839448668634013902747536039084660630117958943055290547871647862562244221739885713244813289
20
9
10
$\log D_0$:
-0.3702041019728575804747239785034084233883221003438921640550788231061140906207950055282035075847160710
Definition
$D_0(p,q,u)$ is the zeta-regularized determinant of the minimal scalar Laplacian on the flat complex line with character $\chi_u$ over the unit-radius round lens space $L(p,q)$, with zero eigenvalues omitted [6]. 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)
$Z_0(s)=\sum_{m=2}^{\infty}a_{p,q,u}(m)(m^2-1)^{-s}$ for $\Re s>3/2$, with $D_0=\exp(-Z_0^\prime(0))$ and $\log D_0=-Z_0^\prime(0)$. The spherical-harmonic degree is $m-1$, and $a_{p,q,u}(m)$ is the complex multiplicity [1], equation (6).
(2)
Let $W_j=\{-j,-j+2,\ldots,j\}$. Then $a_{p,q,u}(m)$ is the number of pairs $(a,b)\in W_{m-1}\times W_{m-1}$ satisfying $((1+q)a+(1-q)b)/2\equiv u\pmod p$. The numerator is even, and this is the complex-character projector of [1], equations (7)–(9).
(3)
$\begin{aligned}[t]D_0(p,q,u)&=D_0(p,q,(-u)\bmod p)\\&=D_0(p,p-q,u)\\&=D_0(p,r,ur\bmod p),\end{aligned}$ where $1\le r<p$ and $qr\equiv1\pmod p$.
(4)
$D_1=D_0D_\perp$, where $D_1=\det{}^\prime\Delta_1$ is the full one-form determinant and $D_\perp$ is the coexact one-form determinant on the same flat line. The nonzero spectra on exact one-forms and on three-forms equal the nonzero scalar spectrum [1], Section 3.
(5)
$T(p,q,u)=D_0/\sqrt{D_\perp}$. For $u\ne0$, the Ray–Singer torsion is $4\sin(\pi u/p)\sin(\pi v/p)$, where $v$ is the least positive residue of $ur$ modulo $p$ and $r=q^{-1}\bmod p$. For $u=0$, $T(p,q,0)=2\pi^2/p^2$ at unit radius [3], equation (4.3) and Proposition 5.1.
(6)
$\displaystyle\sum_{u=0}^{p-1}\log D_0(p,q,u)=\log\pi+\frac{\zeta(3)}{2\pi^2}$. The sum equals the scalar logarithmic determinant on the unit three-sphere, whose multiplicities are $m^2$ [5], Corollary 2.9. Here $\zeta$ is the Riemann zeta function.
(7)
$\displaystyle\log D_0(2,1,1)=\log2-\frac{3\zeta(3)}{2\pi^2}$ [2], equation (1.1), and $\displaystyle\log D_0(2,1,0)=\log(\pi/2)+\frac{2\zeta(3)}{\pi^2}$ by (6).
Comments
(8)
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 scalar sections satisfy $g^*f=\zeta_p^u f$.
(9)
For the flat connection with character $\chi_u$, write $d_u$ for the exterior derivative and $d_u^*$ for its metric adjoint. The nonnegative scalar operator is $\Delta_0=d_u^*d_u$, with no mass term or curvature shift and no rescaling to unit volume. The prime on its determinant omits the single constant mode when $u=0$; nontrivial characters have no scalar zero mode [1], Section 3.
(10)
Each complex eigenspace dimension is counted once. Realifying the complex line doubles the multiplicities and squares the determinant, so the extra real-dimension factor discussed in [1], Section 3, is not included.
(11)
The spectral zeta function $Z_0(s)$ is continued meromorphically from $\Re s>3/2$ and is regular at zero. All eigenvalues in its defining sum are positive. The prime on $Z_0$ differentiates with respect to $s$, and the natural logarithm of the positive determinant is real.
(12)
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 (3). Conjugate characters have equal spectra, and residual automorphisms can give equal values in distinct rows.
(13)
The coexact one-form determinant and Ray–Singer torsion use the same quotient generator and complex character convention. (5) relates them to $D_0$. The APS eta invariants measure spectral asymmetry of the odd signature operator on these bundles; their presentation selection retains orientation.
(14)
The case $(p,q,u)=(2,1,0)$ is the ordinary scalar determinant on real projective three-space and repeats that determinant and its logarithm. The case $(2,1,1)$ is the nontrivial flat line over the same space.
(15)
The scalar multiplicities are quadratic quasipolynomials. Their residue-class decomposition, together with the factorization $m^2-1=(m-1)(m+1)$, gives a finite expression in the Hurwitz zeta function and its first derivative. The attached continuation note proves that this factorization has no multiplicative anomaly for these spectra.
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)
[5]
J. Cunha and P. Freitas, Recurrence formulae for spectral determinants (2024), Corollary 2.9 and Section 3.2. (arXiv)
Links
Similar tables
Coexact one-form Laplacian determinants on lens spaces —   The complementary one-form spectrum on the same bundles; $D_1=D_0D_\perp$ and $T=D_0/\sqrt{D_\perp}$.
Ray–Singer torsion of lens spaces —   The nontrivial-character torsions relate the scalar and coexact determinants on the same marked quotients.
Spectral determinants of Laplacians on spheres —   The sum of the lens-space character logarithms is the radius-one three-sphere scalar logarithmic determinant.
Spectral determinants of the Laplacian on real projective spaces —   The ordinary three-dimensional member is the scalar case $(p,q,u)=(2,1,0)$.
APS eta invariants of lens spaces —   The spectral asymmetry of the odd signature operator uses the same flat characters and keeps oriented presentations.
One-form Laplacian determinants on lens spaces —   The full one-form operator adds the coexact spectrum to the nonzero exact spectrum, whose determinant is $D_0$. On the same flat line, $D_1=D_0D_\perp$ and $\log D_1=\log D_0+\log D_\perp$, where $D_\perp$ is the coexact one-form determinant.
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 $L(p,1)$)
Repeats values from: Spectral_determinants_of_the_Laplacian_on_real_projective_spaces
How they were obtained:

Exact weight counts give scalar multiplicity quasipolynomials. The attached continuation note proves the finite Hurwitz-zeta formula, including the vanishing even pole that permits factorization of the eigenvalues. Arb evaluates the finite sums, logarithms and exponentials at 513 working bits. All 664 values retain at least 475 bits of relative accuracy, overlap recomputations at 150 requested decimal digits, and are stored with 100 significant digits. Each stored decimal interval encloses its computed ball.

more

An independent 175-digit mpmath calculation uses the coexact spectrum, trigonometric character traces and known Ray–Singer torsion. Its largest absolute discrepancy in the scalar logarithms is $4.46\times10^{-159}$. This floating comparison corroborates the values; the Arb enclosures justify the precision. Nonzero closed-form controls precede the full comparisons. Exact symmetry tests cover all 1744 unreduced labels through $p=20$, and character sums recover the sphere multiplicities. Independent harmonic-monomial counts test 3320 scalar multiplicities. The published sphere and real-projective determinants, nontrivial torsions and the existing coexact companion supply additional controls. The attached checks also compare the stored decimal intervals and test the covering identity on the read-back logarithms.



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