Values of the Epstein zeta function of the classical lattices

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Numbers

family

$n$

$s$

$Z_L(s)$

$\mathbb{Z}^n$

3.289868133696452872944830333292050378437899802413596875471116458740014940806401747667257801239517411

$\mathbb{Z}^n$

3/2:

2.404113806319188570799476323022899981529972584680997763584543110683676411572626180372911747218670516

$\mathbb{Z}^n$

2.164646467422276383032007393082335805549501903837453815365952430888241232373937693113819271883399834

$\mathbb{Z}^n$

5/2:

2.073855510286739852662730972914068336114161839003825623948385355807607179572562969120086213114266673

$\mathbb{Z}^n$

2.034686123968898279429035859581841055803634980065707123684817328008664365803915795765547955877070341

$\mathbb{Z}^n$

7/2:

2.016698554763845653679595099699593519199727121130477412834566273143202956634711470692193937827702648

$\mathbb{Z}^n$

2.000492173106616096597275996095479341920832176916006809066081904266504039363881826098085617103801399

$\mathbb{Z}^n$

3/2:

9.033621683100950305730515279317058539503165384574397544494598772150364679088249540614595413835943186

$\mathbb{Z}^n$

6.026812039691940123546260192728285583941790807253724123978982398916865203781425041876494663093841843

$\mathbb{Z}^n$

5/2:

5.090258233665482945657401531942041885190736305970430927672483383399700081150328063446675846386556047

$\mathbb{Z}^n$

4.658913615603843440161123907680531538588289452283229209212733986040557343051300954916403532302185067

$\mathbb{Z}^n$

7/2:

4.423117787678166000664268648432545590978451952397697849565222149339145573652864731561773339233824729

$\mathbb{Z}^n$

4.281430660805780585620776865437441599012329813656095658555535345448793598139008744913939423480499891

$\mathbb{Z}^n$

4.064021927721303484831337288457069188191372152176785779603072752065071904668978287790459629348617809

$\mathbb{Z}^n$

16.53231595976166964389270459288785174383412907025518688611779365956027030949510852307782420104611771

$\mathbb{Z}^n$

5/2:

10.37752483084708386472894835510703227067986136431381719203634962250422225082307091630043911391141478

$\mathbb{Z}^n$

8.401923974827539993146138987215054975272768952161665274396448009611145311306704220274765886663675835

$\mathbb{Z}^n$

7/2:

7.467057780918810530923028822536586990080803778876475907793288069069510818366206871830028176768984285

$\mathbb{Z}^n$

6.945807927226369624170778023111715164374908557886209347710241852676026500179846589773576018121904648

$\mathbb{Z}^n$

9/2:

6.628859198886779099036097213251052364532963796059547965660540122425625077295559159573053333607380460

$\mathbb{Z}^n$

6.202149045047518551930416392285138176011966250498231174646386033199675201599031032645868869415858480

$\mathbb{Z}^n$

5/2:

24.53127768992602917867690280522154520042540324852377327045901705520710412510102685857225471448656218

$\mathbb{Z}^n$

14.82978262722972088647814081113844179061613599995100656521866707791036275609953155776788279391312206

$\mathbb{Z}^n$

7/2:

11.71411822122129794546616898420796458282376816946134924246886563194543688924447505713442972772586280

$\mathbb{Z}^n$

10.24548615194334752146598799695758012161737502103393746126824766340763387843870300827627386952757309

$\mathbb{Z}^n$

9/2:

9.432723962878258561178260832456195230945374013887952171300637712346931079156032659679281283463463732

$\mathbb{Z}^n$

8.943256414789339957980372895384682443222465863347681968081213202607779152185928398802407862142325734

$\mathbb{Z}^n$

8.431048565594719408625292300696913783740098102486207806730577357871062250507423083001365011664432285

$\mathbb{Z}^n$

31.71423334832768682381531497373150504520315601264849447378087901855849392181572926960150626730518461

$\mathbb{Z}^n$

7/2:

18.83607776788864460859401285741880582921952612776682910878093165462652650233216517501392885480760202

$\mathbb{Z}^n$

14.73684665731354710169745725017246645326294406272570011068553995647265694231589013806831536142488704

$\mathbb{Z}^n$

9/2:

12.82701942603694617518527555337708483610917546498738241069714599055068901260837112250628100711356725

$\mathbb{Z}^n$

11.78346412874461109280537171542475569942520147031327780567856676331365725268321730977789142600620532

$\mathbb{Z}^n$

11/2:

11.16284079631792625631044486986806484365698412126277564917853450847004239915849423989337753208119818

$\mathbb{Z}^n$

10.77453537323627540041988536611644486098665213493165355154012920837445247628333853187330331147307117

$\mathbb{Z}^n$

7/2:

37.12440996917590539180717253450619198012724564819646159626228232799068833367209429030530956042774548

$\mathbb{Z}^n$

22.06249395869415598379249883299508802666356117279384032422860309907152729927098128714741549182586012

$\mathbb{Z}^n$

9/2:

17.32149708499633353848283840946987567320497279908607889269216645209756376745916130862162901781463578

$\mathbb{Z}^n$

15.14010569284232321500471173086849867523340352666235235789470985691526132863910407426302481285331794

$\mathbb{Z}^n$

11/2:

13.96260774953247994489842854831415269944638260036453013744330462040001838090379137263954483069033830

$\mathbb{Z}^n$

13.27001999313124718315370897026679074635468617725781913746671149518341207123913389619258065028110359

$\mathbb{Z}^n$

40.36420812560887011032796698079289008552388583073614866631090075118647388431193107251066451078079163

$\mathbb{Z}^n$

9/2:

24.41924637140193566516596692161246791070694796654259487031856224324666922896513478706647895438006183

$\mathbb{Z}^n$

19.45203016885713435666390258893394266649604961245373454261494155353120813261300834966433469856312207

$\mathbb{Z}^n$

11/2:

17.19057370773304726463167129813125085088712701763396406433795156153352667531349269165517194244494691

$\mathbb{Z}^n$

15.98145710321194638723528567918392293206312372399994541755218451083438513068407361667753294882724417

$\mathbb{Z}^n$

13/2:

15.27599457120545294909291231271027955757183876409435884638969026835329769553519192100145567115628921

$\mathbb{Z}^n$

9/2:

41.56574519462492853996359821587037534501116085976703318167201501466125892611151836490347233747938799

$\mathbb{Z}^n$

26.01158628676072796742637230107562079004152005705882555015573258173747274583236799491802450428536342

$\mathbb{Z}^n$

11/2:

21.20421127709639297020650209712165133806182555017566919417615466645882717599413254318857401200258260

$\mathbb{Z}^n$

19.03144739902760002509245724943233129216590496709364683969906201798662115734920774275428316813434320

$\mathbb{Z}^n$

13/2:

17.87667105123848224910304704700909030564764864758194566229367459624476058622499795426040883410636460

$\mathbb{Z}^n$

17.20597014471482501213623460274149593017337882364490444099503795240828380632947350351003073911519996

$A_n$

1.644934066848226436472415166646025189218949901206798437735558229370007470403200873833628900619758705

$A_n$

3/2:

0.8499825875962501838883719729758278688624747660515768504463368923293441015273575791780952772921750525

$A_n$

0.5411616168555690957580018482705839513873754759593634538414881077220603080934844232784548179708499586

$A_n$

5/2:

0.3666093236312104039024497788977841880021589510682330879382703841102980406591955441998478522544831950

$A_n$

0.2543357654961122849286294824477301319754543725082133904606021660010830457254894744706934944846337926

$A_n$

7/2:

0.1782526529603281547760219191857660341563812022856646526687623498055412670566093734236092745639981212

$A_n$

0.03125769020479087650933243743899186471751300276431260639165752975416412561506065353278258776724689686

$A_n$

3/2:

3.901170243481159305813825985174954735432817661573913176088534486293936003213420875386798844626772891

$A_n$

1.927786433226224104376267698049361905953767595543381865926990452299489449343040384857005023265535611

$A_n$

5/2:

1.195346075074407817859835840091997535332308239864691231125957229102943684395360942216641802414299669

$A_n$

0.7969851941037308633338878847310488087108244312307764269731876943074750935268582487293045489409999909

$A_n$

7/2:

0.5475872377559187415940182650296527915419277739459338543886576138479310708959108671230227398753340121

$A_n$

0.3815293630119161384056576036285990936658013672861341263238860023040110138333555896764885850285263292

$A_n$

0.09390334262447043984671761359329168328710648226217362757349494883095902400858982968635273191599546404

$A_n$

6.334576076282542078734800420375778269531181134122837033677876944456729160801023444951019200514623892

$A_n$

5/2:

2.999461840456821380897621519509787204018519316865292981839383560600662687734607747075183000986027238

$A_n$

1.806740130468058982852604583511163523034596465131229468488101315927314933071073591712417865989433693

$A_n$

7/2:

1.180814204461860882714305626069783156565865739841959800209812063676463367159549924549621461975968464

$A_n$

0.8001210769611332847370740436560503016411530007793509793924484252015314833215302361355041905473499740

$A_n$

9/2:

0.5520977804606942636595956328218962112774058986685771326903860471871840055557713394691854555679450082

$A_n$

0.1895606280710090579415929126645689761887837422507438655483483102747879153352460782784058673856511674

$A_n$

5/2:

8.456478538118636032256024430344870396891426532596721102860738710537820331462276474294767939404194435

$A_n$

3.863591004281546514366119682764984337680786241456267244602284009185336027291438708658123244366528542

$A_n$

7/2:

2.269026958280534598181246120734848965063412341158286273189692569006824089154274605875263504420225410

$A_n$

1.456369629338986078105730710744522531674421177078247425475346194080584793965882175470600025163489908

$A_n$

9/2:

0.9741411840816292921134401130690427357233070534742014350617160730381390189782454211197172015016466379

$A_n$

0.6659642581652210366060574190065652044622808293495210520528793635196062005770157560488080303036622352

$A_n$

0.3215175372566409032695316649459104749096107768853008505015252041385328108025445667263991520773082898

$A_n$

9.897114535228656414999032748788490487347768355173657317119969098711738214411027180932472975672553475

$A_n$

7/2:

4.397214202480978807051169428420183667091218570705419998848122034171001976983996178407617184365531420

$A_n$

2.531263353584239493565636516784211246853178712481627532129410149878708460578922538177930034690749650

$A_n$

9/2:

1.601528914606050576179044633259885571189260637558135047209630266149350274162937824371479152026836181

$A_n$

1.060256729297160391751273283760285496678881222704971592831192361462233840497860998661830363039156656

$A_n$

11/2:

0.7195020226325692919582460302161965514879890580142329967884625563737998827550660352541014710783704845

$A_n$

0.4954291860116655226715905749562545600420719849247584465438905268211409593968543944324464077368054301

$A_n$

7/2:

10.49227670588736896902333025850712798240858993726138595057443099745270892616512828902932087788763581

$A_n$

4.568912701251905041275953133297272139905891851045807529014539771683813257788932831384660073439250611

$A_n$

9/2:

2.592991886332499552301914758017986080054811322690457681070563664058323552266199919317338950395459446

$A_n$

1.624183160381974700372206173827617071379959594806010605904024358661246086691588273665484027425601490

$A_n$

11/2:

1.067646952668827070942476344370930520632791515646875009660500154918724076931291187210965979831562797

$A_n$

0.7208930502486073373448603994824343789877306072614359799017537243875720574651807479277630119910213777

$A_n$

10.28043129105528606851738120966033652740628360638010913631372037379033157373914139815921821101631639

$A_n$

9/2:

4.421818762119115134174992468669644048090326769215296172944888812295030773595849671713667733771965988

$A_n$

2.488637057797844059917232557387975335073940051426953127703114874281286609059075195581815735990109875

$A_n$

11/2:

1.550020071930079988904173992768522462230044992231760561419671532253620090994723124026885336828348244

$A_n$

1.014992000260422875822185850000442696509283679475885614216823269864457560330781842542929397767489730

$A_n$

13/2:

0.6835535364882827253450968268830850854979565458182313540464968976917220110003256240099169414055216615

$A_n$

9/2:

9.440707963829915388828834664378081498771148730120258636606068197726494431311644705358255172932792058

$A_n$

4.042045649421810523311566783914871483935942757756797602473809291305011369612007033343871877882133248

$A_n$

11/2:

2.268974181426027755756962845857003627586503408250083791044497025175526699397684271740985170182866668

$A_n$

1.411164485662261844978859477321881597124809619794117883053607593936981004148116099786877179364850073

$A_n$

13/2:

0.9233475848218585270272340623851494510425471440736955026221228094313909504018969988862086177734149522

$A_n$

0.6215850865587397957694633082669649671281083471756469599618639869064306499223553258937311584053845315

$D_n$

5/2:

9.611497185630387813790660906297657906503077289621243930120197923404793867583364980458063384821442827

$D_n$

4.448934788168916265943442243341532537184840799985301969565600123373108826829859467330364838173936617

$D_n$

7/2:

2.639561671816960796278018376360740267728574496566418785869696589685523125521795042271987730070964480

$D_n$

1.707581025323891253577664666159596686936229170172322910211374610567938979739783834712712311587928849

$D_n$

9/2:

1.149051558714740607805657615789348288671780468555279346504583642853028332642059951242118405219741163

$D_n$

0.7891108601284711727629740790045308038137469879424425265954011649359805134281701528355065760713816824

$D_n$

0.3832294802543054276647860136680415356245499137493730821241171526305028295685192310455165914392923766

$D_n$

12.37443260982567442481052781501898251596333707276476044462044715473536442397779913082190742107312120

$D_n$

7/2:

5.587014429768099953898198661313857082916219483994008001863162211754006823446492612341088199502239648

$D_n$

3.256170422669559345569914861203301683619831764422539047250568213604127681730267637440845866304996097

$D_n$

9/2:

2.079603872606763401539940144186197489574180521145238336157644268072556621019097599928561171904854468

$D_n$

1.386500583976870868712935911929303463481942406640371166232376266351989843671878430612327388675890165

$D_n$

11/2:

0.9458498604293822327889390653877166725269430374804839486509290587442390067904392678353557996646893797

$D_n$

0.6538184940306851612089481804807419545724696711997060684685025519980958191995955213132158168323592986

$D_n$

7/2:

14.15861235134000267919020333163170470333143550925032266510003498023849015635190964186114883972572920

$D_n$

6.259151622567062827540907291317386902205790725365046797904127370701935289266198464328049919859960726

$D_n$

9/2:

3.592642303328789667148262229457856367892816833019767732185119119194374295995244313421187132470779214

$D_n$

2.269285893716120630761427430459684382790371492233732101467864115113886267833751715565463029489286774

$D_n$

11/2:

1.500924546452634053488411959360529220718339134137157914961201813780180062771114141624236607820833622

$D_n$

1.018019226770915685574644414841401600257327633335542584458214196689292626246143419464471562498558965

$D_n$

14.78164663991461193042839023237471627842876536400493364661451003170014636282080216441759960343548644

$D_n$

9/2:

6.435275843777702939437326327357386438233314270907022182452075639236692540022984712713614981332882824

$D_n$

3.653617204815356356687503159185989526788770992589434651946670844928741999583802721282115827761440417

$D_n$

11/2:

2.289907177871328959183910418801894344566983516117195335802053389701756575517552312087996652708854537

$D_n$

1.506190599292323967950334961390741264815713308407482279015010141722548703706647699133722409400246581

$D_n$

13/2:

1.017568749203997966395144134127177886439427394785132458161736469798687205667453417702393947618119766

$D_n$

9/2:

14.32682208763888697959754868042701406240751605594968540810320920981953512466863227479316801220091903

$D_n$

6.183081986197222221765285219108139368124623620120540499627182335003169751058513703710022218231766714

$D_n$

11/2:

3.489647254936306135847313755589480511042759227092691129499200334390052490568120751437179228267857820

$D_n$

2.178298196274243376366004143007315027416579484185417409363145652661119289094186428387538434906942896

$D_n$

13/2:

1.428817980259680695602862215805629077334726506032533773877810540359782298521881992783409496292558216

$D_n$

0.9634661181133673074189269128393404559512347904221378601746478139852012240412440564403535547671048541

$E_n$

7/2:

16.50865026541274302014194736981123739404696884524245172978285603509248033054576682846647837700066993

$E_n$

7.352381610928340211493707657112975385039412555508809909516780231508186226159373664593979375896797440

$E_n$

9/2:

4.243862706599299152168837113736058068655190154823104352796631085152165648928636082838152602317498148

$E_n$

2.691910109764037759508564608071520807340071262964911215596967394180826204492379695657949744683655157

$E_n$

11/2:

1.786018569054949062074517811160370775261268872525281301915126237079513795900761442675863343203913133

$E_n$

1.214182181353675849050763587007319512884910897209358708277952871154013343749528250157592548377506629

$E_n$

21.23591975997180002762671157657338915035421440668061085844505829056729946989517821779815300352699646

$E_n$

9/2:

9.354022544538287317008599389891814426922662049878164185810051539930052938535893609787857048023741017

$E_n$

5.356798983071513796187243123231012033051583466206245794116279744302377097288319232010509692075227288

$E_n$

11/2:

3.378701189288412355667783268920069559952509322614322820021886945256755568706115042250355778991800132

$E_n$

2.232625811530647289946475071194673524306562626251536554363680355781872741337669594748516422566647445

$E_n$

13/2:

1.513399257740580163247371611217468400219109947333896495507668945702423413435653222913009627887528148

$E_n$

9/2:

29.22428381216190556401318797496700062846610664330535141443339847521857950329800253571483673926017030

$E_n$

12.79258341971839080365231424643063317543025576576663551647003241724793741598313180077935631358296562

$E_n$

11/2:

7.293623731995054674323012556788973898044552447785442275027247220183971643174068207917065820363147691

$E_n$

4.585890939524722897612640301068031636666483124600878756553990847707619555987760901868501968225142939

$E_n$

13/2:

3.023574422386958334018861886907502108240078813770457348976210898829813219022604245493146717069999369

$E_n$

2.046299719886797821066747425499484154232711059303655632229340489880073396193792686244998700390311195

$A_n^{*}$

2.515413515977056471696640814813392282879679607261178707562033308547465001534407500450689346072763855

$A_n^{*}$

5/2:

0.9467588175213397882899581311873854020728860862540951321054756377928315266653876049366268714294301625

$A_n^{*}$

0.4538395506404564010703153525259998792468037721544452671630536236095964973070561730090129688366359974

$A_n^{*}$

7/2:

0.2363766832751650769508808668585821558708927832908136945299301845538759456820446017114834237868187705

$A_n^{*}$

0.1278419494864507991002954299945113850760690849018430531780168468760171682454605975320140038732514261

$A_n^{*}$

9/2:

0.07052646915552941514492757484754467487434397972711773003352405493432038802940574074926894667244858636

$A_n^{*}$

0.01250230900970556771969335677435844641157182633026155184480564518318251738133337017848789005564203304

$A_n^{*}$

5/2:

1.133853584710572832971423638710114923021529737359639785681577036023014120591784129017799008412347331

$A_n^{*}$

0.3482650546671114833022926973034331645716892574112902589828299307726327613834878236697620882699277187

$A_n^{*}$

7/2:

0.1379759861758621631368724260190853207541614177242446411945737007843613655287880889836236522043974807

$A_n^{*}$

0.05997474944394216957230140920607204300647322543434976789794647296998889045962948690556248674409653776

$A_n^{*}$

9/2:

0.02728020727332964176735865676624885392058511921690812288781870518600536846645307837366254233368063928

$A_n^{*}$

0.01273594112428926519173784790004171530218156546014691870544878087579421343712752796455547387521818353

$A_n^{*}$

0.002902506758153179811183941585618716123620556435306814947728281479676874373207813887796753043506873318

$A_n^{*}$

0.3959585686315380423771086200420457823529007410561835727685249707319837752514102051901206441063019251

$A_n^{*}$

7/2:

0.1040425367265221192348037089825059306021581734336931876007575700244132811177569577233239468291373735

$A_n^{*}$

0.03565729453589664122453416133621068895118642576616387178950059216608605587660890922454766640107996854

$A_n^{*}$

9/2:

0.01352632540474310869298710968813365782057971860520145126929013864237457692181124882716359790448609709

$A_n^{*}$

0.005406890276072437690453933884707254776015689042634096598312045630903671197435138750252028241007988596

$A_n^{*}$

11/2:

0.002230409478418012235853501696333002086532128633366651009746361424872043555752700655864298757326388152

$A_n^{*}$

0.0009393953560396770833347745607677760851342699042635393483380413386929648330692066921414716897260164187

$A_n^{*}$

7/2:

0.1134892178450558633902902225346060388019568663949226270260745622057554859402394718431553018600859618

$A_n^{*}$

0.02645263240883301527404830671761923965763781851483233292891286209692315343731569860653803835279309320

$A_n^{*}$

9/2:

0.008119882191646234263240707413219355975392359460420227644369224738429995317508213389430969080491817794

$A_n^{*}$

0.002779154719163149817644043800526835273301411734647372385048110589327520172403312456128022473825329353

$A_n^{*}$

11/2:

0.001007757982785980655457175136182150815532702380351891522084948484318388502536772551509476031691359792

$A_n^{*}$

0.0003785460088230269334957936150077948532551814908301363168585080256062938008228884792937927531814183062

$A_n^{*}$

0.02773757057648216504188040408789344452482005215208918674087815750101500470438345195007522286065810714

$A_n^{*}$

9/2:

0.005912824029441585544042555844215316870412920825346520876706511635285992816112594505735779704100380861

$A_n^{*}$

0.001672530695974735158505904343223643291867248313442154434680990077186279463668745893706850731794081137

$A_n^{*}$

11/2:

0.0005301642728775738817959411442423024399089046087974416385350122171816179131295399367225006139980712419

$A_n^{*}$

0.0001785772053304843504308822842105574661699867067272035992737541218479956535487170502814795421742707201

$A_n^{*}$

13/2:

0.00006240481939871305848305056975821003373206179112237406805308923640890691665590859416835815078034502755

$A_n^{*}$

9/2:

0.005946926363755025438133065271836199045284800370343190098897689475984212691002226276997417701950365384

$A_n^{*}$

0.001191726099313031864834642456831962379240915529389851462429431842379711509453751060548084409120199174

$A_n^{*}$

11/2:

0.0003182068326558148275831441854501220498786908130472835235104493056873766910151360616710296345905975968

$A_n^{*}$

0.00009536040809406496488770744770401803643765753347296159950593124064274157870799168660051903340855335450

$A_n^{*}$

13/2:

0.00003036403290883097335878044174531918571292607353518298186499417325275292481356945292146631924313808723

$A_n^{*}$

0.00001002062071859652737205304314447549000701776544947572330860017535742252723750290283998241638206925634

$D_n^{*}$

1.022312813958967525475672389291875279600562637759334718768272895838288172552310168193295796016805690

$D_n^{*}$

7/2:

0.3055768230230641756172048524899045070008859652530215019626612008692608709342739954547588576526960675

$D_n^{*}$

0.1181488335744976094780773377327114151151864177198050062049299760140544182169818862601073474422345445

$D_n^{*}$

9/2:

0.05017908034565260293634646049125965937097488172951866590418548824782396960701853827773742556518743034

$D_n^{*}$

0.02230640920246987717707052477962498561648998186728815620950177243741566590957852516806777007508664587

$D_n^{*}$

11/2:

0.01017485782434277515533412222029425157502300109514558297671798200952171604405049634445651065508894595

$D_n^{*}$

0.004716756027526016728070108822430130430360581006539569116531245887703408526597105508140866506573111167

$D_n^{*}$

7/2:

0.5620534804077728394138718932638514709209712548529449330914677992170237369811419356996980941525105777

$D_n^{*}$

0.1588444348812152293485621484983945265316430352832570758543277744394792479389957969763793553339051405

$D_n^{*}$

9/2:

0.05872599050404658362867252950130882548290417402459559261072845956344245234704294703041843138601083885

$D_n^{*}$

0.02408890519762309289576469437212963380371195220181511361329607500728082869152707400953364937976103850

$D_n^{*}$

11/2:

0.01043505958771086391850302854810794870584502433888221004328895758298846244202315696478736310551430974

$D_n^{*}$

0.004675271298115750152309774615172856700408056065681133914093159717977860454853479043893450201163063346

$D_n^{*}$

0.2877640456848176311139637380892065629230773854237470267576257546939989456187684081970280796438023975

$D_n^{*}$

9/2:

0.07914516597080664113764606898311202273134552867128251142439267491317867170887603613217601239186412065

$D_n^{*}$

0.02881122809458632523746184337071184528579958222918761542940243173997695646357081541823061889917911069

$D_n^{*}$

11/2:

0.01175008946684116530092554134860598734073598537760232407654429673302016798512784259609658593720291299

$D_n^{*}$

0.005099612543099405921291813918945134069964146165560494062465928161152318353423492537624098899319820820

$D_n^{*}$

13/2:

0.002302132572232064592076545980419788831663621237845814730468737504502005848807468033648373444493388712

$D_n^{*}$

9/2:

0.1393763059446698548999899937596686493693912930360905570201804561434752884440825368881773628742146299

$D_n^{*}$

0.03831112222051080579218792026925840664516873471518653865609514916623731257390781659087567645994898557

$D_n^{*}$

11/2:

0.01406843956602240724958881821265656157815693944900447826427355880763997337949513059382243515439119255

$D_n^{*}$

0.005821931856818495866109797257215274538736746154278459358906433693378813889437587082450246639348325997

$D_n^{*}$

13/2:

0.002571555656188114932365117967433207564387006013556859474180966224754006643948540339316392001423492054

$D_n^{*}$

0.001182350912369487673305168190189317830000996787195308836981471148937867929115879563178669496482032022

$E_n^{*}$

7/2:

1.273428078566316128448906760671485119516655204459583028129902342616211451374113145655009335892722262

$E_n^{*}$

0.3941204130869289483382950701645949709219098313977745105070003353464336748748693695076905706297319533

$E_n^{*}$

9/2:

0.1582709606716353003662406963684083437571430904718310758450944334124818928436931327801852411256573467

$E_n^{*}$

0.06993391665429146768979210484228284795424085491548093011976711706654116726414248151698529728483578953

$E_n^{*}$

11/2:

0.03236501012302536468717319933765746618356101459951116341075754706768852512310364440128951688464053447

$E_n^{*}$

0.01536805569557582221278479946577561562369453914517383848052583845551010556763117261212693079267255090

$E_n^{*}$

2.937894621366378818865539700245223045505942746216945532709646763950184180060566020978191567574397660

$E_n^{*}$

9/2:

1.014481077367708723886186042520631271683670123433162316502423043228826986102223922860764284717599008

$E_n^{*}$

0.4563380076782209664979179196266373676017219491854681839766902705131007600363574653303377675826031745

$E_n^{*}$

11/2:

0.2265752408085639009601077904315089940659152593552110201666090023264813163107566722638517503497225422

$E_n^{*}$

0.1181327488682613660678958195245251041185131794876461598367647284337013413182260607740931868974701309

$E_n^{*}$

13/2:

0.06333584794930485144338122259817777740598189083216893728963110359587368400825492709643752292589257658

Definition

For a lattice $L\subset\mathbb{R}^n$ with Gram matrix $G$ in the normalisation of the classical-lattice packing table, the Epstein zeta function [3] is $Z_L(s)=\sum_{0\ne x\in\mathbb{Z}^n}(x^{\mathsf{T}}Gx)^{-s}$, for rational $s>n/2$. Listed are values for $\mathbb{Z}^n$, the root lattices, their duals, the laminated lattices $\Lambda_n$ and the Coxeter-Todd lattice $K_{12}$, using the Gram matrices of the Catalogue of Lattices [4].

Parameters

family

— family of the lattice

$n$

— dimension ($n\geq1$ for $\mathbb{Z}^n$ and $A_n$; $n\geq4$ for $D_n$; $n\in\{6,7,8\}$ for $E_n$; $n\geq3$ for $A_n^{*}$; $n\geq5$ for $D_n^{*}$; $n\in\{6,7\}$ for $E_n^{*}$; $n\geq9$ for $\Lambda_n$; $n=12$ for $K_n$)

$s$

— exponent ($s\in\frac12\mathbb{Z}$ and $s>n/2$)

Formulas

(1)

If $\widetilde Z_L(t)=\sum_{0\ne x\in L}|x|^{-t}$ is the convention using the length exponent, then $\widetilde Z_L(t)=Z_L(t/2)$.

(2)

For $s>1$, $Z_{\mathbb{Z}^2}(s)=4\zeta(s)\beta(s)$, where $\beta(s)$ is the Dirichlet beta function.

(3)

For $s>1$, $Z_{A_2}(s)=2^{-s}\,6\zeta(s)L(s,\chi_{-3})$.

(4)

For $s>4$, $Z_{E_8}(s)=240\,2^{-s}\zeta(s)\zeta(s-3)$, because the theta series of $E_8$ is the Eisenstein series $E_4$.

Comments

(5)

The root lattices use minimal norm $2$, while $\mathbb{Z}^n$ uses minimal norm $1$. The value is scale-dependent: if $cL$ means all lengths are multiplied by $c$, then $Z_{cL}(s)=c^{-2s}Z_L(s)$.

(6)

In dimension $3$, $A_3$ is the face-centred cubic lattice and $A_3^{*}$ is the body-centred cubic lattice in the physics convention. These names refer to the same rows indexed here by family and $n$.

(7)

This table contains values in the absolutely convergent range $s>n/2$. The meromorphic continuation, including the pole at $s=n/2$ and the value at $s=0$, is not listed here.

(8)

If $\mu$ is the minimal norm of the Gram matrix used here, the Lennard-Jones lattice constant in nearest-neighbour scale is $A_{2s}(L)=\mu^s Z_L(s)$ [1]. Thus $A_6$ and $A_{12}$ are obtained from the rows with $s=3$ and $s=6$ wherever those rows converge.

References

[1]

Jonathan M. Borwein, M. L. Glasser, R. C. McPhedran, J. G. Wan and I. J. Zucker, Lattice Sums Then and Now, Encyclopedia of Mathematics and its Applications 150, Cambridge University Press, 2013.

[2]

Audrey Terras, Harmonic Analysis on Symmetric Spaces and Applications II, Springer, 1988.

Links

[3]

Wikipedia: Epstein zeta function

[4]

G. Nebe and N. J. A. Sloane, A Catalogue of Lattices

Similar tables

Packing densities and Hermite numbers of the classical lattices — the same lattice families, dimensions and Gram normalisations

Covering radii and covering densities of the classical lattices — another geometric table using the same classical-lattice keys

Kissing numbers $\tau_n$ — lists kissing numbers of the optimal lattices, and its comments link the classical lattices that attain known cases

Watson integrals of the cubic lattices — another lattice sum for $\mathbb{Z}^3$, $A_3$ and $A_3^{*}$, integrated over the dual torus instead of summed over nonzero lattice vectors

Values of the Riemann zeta function at rational numbers — supplies the $\zeta$ factors in the closed forms for $\mathbb{Z}^2$, $A_2$ and $E_8$

Values of Dirichlet L-functions at positive integers — contains the Dirichlet $L$-values that appear in the closed form for $A_2$

$q$-expansion of the Eisenstein series $E_4$ — its coefficients are the theta-series coefficients used in the closed form for $E_8$

Data properties

Entries are of type: real number

How well the digits are known: proven

Table is complete: no (it holds every lattice of the classical-lattice packing table, with every integer or half-integer $s$ satisfying $n/2<s\leq n/2+3$, together with the rows $s=3$ for $n\leq5$ and $s=6$ for $n\leq11$)

How they were obtained:

Each entry is computed in arb ball arithmetic from the Epstein-Terras incomplete-gamma expansion [2], with pari(G).qfrep(B, 0) used to count one vector from each pair $\pm x$ of vectors of norm $x^{\mathsf{T}}Gx=m$ and a factor of $2$ applied to the contribution.

The same expansion is evaluated for the dual Gram matrix, and the truncation bounds use the explicit positive tails of the upper incomplete-gamma series. The generator checks the $\mathbb{Z}^2$, $A_2$ and $E_8$ closed forms above wherever their parameters appear, and checks the Lennard-Jones values for $\mathbb{Z}^3$, $A_3$ and $A_3^{*}$ at $s=3$ and $s=6$ against a direct summation over lattice vectors at lower precision.