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Zeros of cross-products of Bessel functions
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Numbers
$\nu$
$\lambda$
$k$ 
$\kappa_{\nu,k}(\lambda)$
0
5/4
1:
12.55847031498140748520907583136666352116115776362692470101565756694427717818672558963928396617395422
0
5/4
2:
25.12876967977675457898540722499413657025104494194973524675752253522877612375319774970771959499340938
0
5/4
3:
37.69646143922955584164727031384606492676901788492054278597230727466671495469429612661270744773626292
0
5/4
4:
50.26349394095335288900781453289531559356917059609111752866332590919253526950030633025552524582119689
0
5/4
5:
62.83026199385516905790975877363622671528288139714227724500600356972485349973655386832938291751099969
0
5/4
6:
75.39689766790832336065531263532177676200354807768443293414070477457066744776020444512738417431938190
0
5/4
7:
87.96345765141150385454903376771498682226950377805081839405133583862280872356881352728666343542062072
0
5/4
8:
100.5299703116669250008971447736978724879371070077445941590005337335753149306285159966300977452573368
0
5/4
9:
113.0964514160117651971377998993403493305677446241866570817200175878518609198225573070439224575119767
0
5/4
10:
125.6629104278616834516129245543077344238640364683191217995849045642037366100990566577177221563841442
0
3/2
1:
6.270235215795336893206450234539677846992276486579039910286027529024219576837303778349250905938148167
0
3/2
2:
12.55978081885759972758310928505201539674244732196040382339308995000523871295588038277204113732436917
0
3/2
3:
18.84514745792063838466191898378350195386323732860775324246752842712721921674081109480867826734653491
0
3/2
4:
25.12943080214962013826832359819203975490671638711822538329953593973839012439875352509714543085832632
0
3/2
5:
31.41327667385198261440408192894249717421500198319930512787789004149724758539639783006189875142463232
0
3/2
6:
37.69690293393208902901038667932272873058173000710019833675716525084725357195997288677099732490264239
0
3/2
7:
43.98040344185691443093729806780656075943821004643969731842371038768251364619415382958897420482867458
0
3/2
8:
50.26382525919843030743140885635942976191696534191205677342560154580708724686984506194440091908992978
0
3/2
9:
56.54719457546765468205154825229704638852223718642759957625469486523702466196707011468563258763655054
0
3/2
10:
62.83052712165970596258194082884895536505675241622523294699005188046565878810279405183463773217898273
0
2
1:
3.123030919595692205078465748218145438304251329564707797527152754704441046662802309032711795323143293
0
2
2:
6.273435713992180653201778289952375313871125474705962542527554181139203654114053892163629026573768109
0
2
3:
9.418207542251576959760624768677271606378416255306613728971166549892950457148835774071006312682154171
0
2
4:
12.56142318552536311093628252022558337766695001631697402247313432449398345543805728195577366368192250
0
2
5:
15.70399789274403760469678158410146306638687376396243402533978368495869778314863309000043607540315179
0
2
6:
18.84624803828837906591202425952327328303316795717878243188105752780533478137005406679770707380472312
0
2
7:
21.98831147548162526788846101866814861572696029060919188471288913447712915685822421225869814369177931
0
2
8:
25.13025775640670071480292529026041311922127713441084934827917558455054430664984217455557858442878040
0
2
9:
28.27212573402984503095443875738941024524220274365970467473907575502765157905113951035449638228306160
0
2
10:
31.41393880423784791144265060354243297724631010465459131891580853137358458458565536639494456852424575
0
5/2
1:
2.073228849054941489470725827852710123597756883116843943889646894147670921865192189207820303802496350
0
5/2
2:
4.177298827521485650752403386219460654440329535879287758194154318178904876611266893876253256309854134
0
5/2
3:
6.275369985464413676150575339023534830381036322999899109642798383545094957372415944883655896652492729
0
5/2
4:
8.371674084049339299134081091344593280229027371726948819562247535042405875872630690009819539971561995
0
5/2
5:
10.46723309437922318892144586754689449177704564516238166254615118293050533184717923078232105212668346
0
5/2
6:
12.56241055638481566205284302664916798368360488270773136907873198345541561519210470528207262572171463
0
5/2
7:
14.65736716661980229797379211970901397826736916843013168071262495639909430177395686013990447682122631
0
5/2
8:
16.75218467309767151878988699148466102330431643397728393510489249015015087642173013392522103809472077
0
5/2
9:
18.84690897808562192027835072153204133210083349094060280452912669447118856465079072553480394897659898
0
5/2
10:
20.94156781786353620397748543020846107877289331715660747881152350717646211855446996998499808175526586
0
3
1:
1.548458778289445551273737851225865973861088259587945857343167301097601713495611274288763826120432669
0
3
2:
3.129084015718066913250505106732191238627822975179185749919077543842256304773276922774908068202430927
0
3
3:
4.703797206969749867774144545549923566470666160212216392635580937191351195795445451592808358428911882
0
3
4:
6.276664632404236345706005109808423084334187210987848825402647495842283245577756091877909645934224442
0
3
5:
7.848734619324136435273069860230452251296969254801085664765901846980617010798246671875231641991096793
0
3
6:
9.420391128626720223724963617808519081448375279200628711352760007235015216209445568295834323631046225
0
3
7:
10.99180657660129337520904452601264690764893128374765975568704674590527675736181819816053599634623120
0
3
8:
12.56306950580786891645009763062806403979130640573725601233448707686835675413962431435817048593876566
0
3
9:
14.13422993801537982392982385468014709377863210367531982714987354993652483613322520062065728613986582
0
3
10:
15.70531822475512099051282499970773576187760852910443235589872536408245392253894183560833316751038980
0
4
1:
1.024421384819211034165362786022334251052291547779479340867360187247017334722798634944673330434683108
0
4
2:
2.080943409484137652013155634645934389491404977839745315503002435862900140382696610082244818098304826
0
4
3:
3.132166790427544854744726854198960021743093131381517008577360771959062648825401684273723428437525741
0
4
4:
4.181569189457428702627494312808188220561835049650064644692250438967549530698900623911623058213324204
0
4
5:
5.230147795043513930196411375442579878715086760530649413688664629003544531893860525503625618460808158
0
4
6:
6.278288080245396247164591159596240130688445665498831419419979023823517076359268007952359073917419058
0
4
7:
7.326168736450981602735583411985587975156426334954860396854413303438835133316215491987557872469664700
0
4
8:
8.373883413477679475949844749343329925534208126331575904661877283569861720686174304701074902853395765
0
4
9:
9.421485750125928289121846162021443233333038067609157755788257376220464354945847695074510300120327421
0
4
10:
10.46900860933137725676381656406608667459008210953505371905464605566690510065726910333842732132812777
0
5
1:
0.7631912660914152788764654110282296586352626660953765364367997305452653187035422828665086603163545197
0
5
2:
1.557106340647199491156021563796339870024656934692602282050079561856941734372002128748163981392659158
0
5
3:
2.346420710633936657569261856519433241484594776198518979457077327317639409053174811896568307275657247
0
5
4:
3.134032424168111287195001899845224762689769144100857507511236630072097025450506675805511150268197703
0
5
5:
3.920842458265441332884130700377770022067362441169608510178219883800956994471388823557400962116093195
0
5
6:
4.707215322506552013968919009596233638549324584252479893822719397586663107628326605232735029760553319
0
5
7:
5.493325023049526934429632569845078700592290950573322939893056068120576278322913447787347325893958597
0
5
8:
6.279264603596390075369839660773647411795883695079269543421026046983451924157373323249899582657838327
0
5
9:
7.065088123078782442346674564357361186520825639152173503088677811220960929669330428116465696798268142
0
5
10:
7.850829051448395774829208732272048312321959823217796483026479977634180154531362910000971817484694124
0
10
1:
0.3313938715053228329292382470411018898233796372980965710728398535348292184857551574456702763601288144
0
10
2:
0.6857579828347392793360699476902136260485378297396647875207782576025031842903785733539489460794609108
0
10
3:
1.037742046297300823470518561516391427522604264551360283894292242819533769805472803761605666852554186
0
10
4:
1.388642141216831508913567998224392327134702733060973015020148133947559566357911786945405118283483219
0
10
5:
1.738963169812845522374713099533181901654747611722991848791477308740009666056767968729435072759315400
0
10
6:
2.088939259587958212591565864050406327613794497317525435674631574509052848764639816766284406715580422
0
10
7:
2.438693767742553878230994134111591301674970797641106231922404621705619971175091459360917375674309977
0
10
8:
2.788297806790888845816753887744591185554833891027484891507752485324259393639812736186832786251611674
0
10
9:
3.137795198790285701733077667301739129887676649975867903386661029750172630598673182570362629716782019
0
10
10:
3.487214379600421852739857272870433112174441044376939003001315310311409379443659612805998282154579688
1
5/4
1:
12.59004151155316942719447334151402595134816368824621012480684951936590657244038504306168405123207771
1
5/4
2:
25.14465214889977301391104853499173226725981562745555460401404874113174129340816177457892904123826701
1
5/4
3:
37.70706195205497578493870086732820034449876910047575230857216408125116885530494557659461685007174032
1
5/4
4:
50.27144754199314591544915325603609887694328025056046087586098710462106724086459243512084141129858442
1
5/4
5:
62.83662606765317964370009445828717491456193543015263469918343785968549972856901336937479635118825601
1
5/4
6:
75.40220160320334732888919527254579918760499213910458219851794575308206368523766080374694953461394378
1
5/4
7:
87.96800416111995561025643609650658719922484222261342139012875501181568983295964175746812594179551519
1
5/4
8:
100.5339486664121474949600929790837894552666124443547125939716563483848089577870534005225081155260904
1
5/4
9:
113.0999878281046074714661024018743572474830243761951156241857367743727404944799824640101812661773108
1
5/4
10:
125.6660932610465299178745647890497511619165214672022306133364590371767849400694471610062762180509967
1
3/2
1:
6.321871910549067735214621666681075541033749906185840005363537405657165069037659668539502341246130075
1
3/2
2:
12.58611991021753700667125858545350199799957935791418849466589013602857635677010142630759655734461907
1
3/2
3:
18.86277539889735452870475059475471845268607439250112907991178828680831005236528243650137100981906128
1
3/2
4:
25.14267002032001716877071157541614488382718943399226214006119979810143801003499081558735867049463860
1
3/2
5:
31.42387484978685231554993291099067275525486502947224703342056624130081832947700820667563150734725377
1
3/2
6:
37.70573783479413547452424914616140443849975150228609176021577635003544363172435560452890607409196378
1
3/2
7:
43.98797781235361385306259428278108039235722512831336218587432487743271754388523928197054198700926431
1
3/2
8:
50.27045374199843218197411640134673342338867035237380721118044397564527295719054580910813558337546144
1
3/2
9:
56.55308711427558933939175459735101336458729774040417688501495257360605770755894834582965703085858295
1
3/2
10:
62.83583076346089231097857425149687725098065989298014546212591578513303458985867483566461903882129332
1
2
1:
3.196578380810635005395253974836030650385521232984813841587886659545694815421399414085436016519228656
1
2
2:
6.312349510373263126551456813601100530982698227680445472438784258957267676852518473325647694272356841
1
2
3:
9.444464925482272759005621507210798338413765835021042042219890027905614910078826087437393895308788794
1
2
4:
12.58120281010410843077550005874724103977930778504962268391541014810739884254799788768564582492148507
1
2
5:
15.71985426942973884548953437618197834638233707425872320470557858034818732205451128424555139193823775
1
2
6:
18.85947662013839320238668932672133401020212776724620002631172005184728965584327367186961147813423733
1
2
7:
21.99965802121732583385269630528195739016539507500443898760281247688937108752134498126604960568621808
1
2
8:
25.14019040687955532886494936020640348198881096113243703156892256150532327032741109596182828237149502
1
2
9:
28.28095745833174696500868878745633675457803761019015174328986190931986267402993844554521159646179366
1
2
10:
31.42188909815750837892532848131320641232598128206342910604971048935877254568956763618178668263573436
1
5/2
1:
2.156472479826978419459394651229655167973705780644489430391871588176449553528856544737373742633050934
1
5/2
2:
4.223090582076393048930199736286804881686240238652727980590153860598434884210234656715215727175915041
1
5/2
3:
6.306583272148468095011964913743163878978620085203367650055964291977062737482938510951207340597102572
1
5/2
4:
8.395279984307007483457457535885806187946005466477132813469029263887860331122950583248224038183159659
1
5/2
5:
10.48619307911305206647277407066025524185767413820172561038951414163054902924229109415411060525422908
1
5/2
6:
12.57824527023023410521549156556412417836899724099636114097266571342256185755828646610439921435405923
1
5/2
7:
14.67095792748187221806009666465342297516091686767036136147539022957838370854071767117871703957858420
1
5/2
8:
16.76408696895388743594630056612680742708043834609734083657442747405782040777745803476587968371553614
1
5/2
9:
18.85749515263560416827853115247206075405281243441290169058184104904202231586469480075088711516021784
1
5/2
10:
20.95109948048059093898895650608147890483099929860655338473330105119008532996760242303369257421015594
1
3
1:
1.635615999904734098146519066928378432865193922612075220799567122780890640542967246117368002846110696
1
3
2:
3.178840674409228151369528548437527631615082941076146675400377643262444795430831689649921981739756959
1
3
3:
4.738090644722869379982201147572625903492552910196005451528995989546314455105892193160564540439911677
1
3
4:
6.302718750111366329859294783523521765072746508881050064166061310611218281811202246099261067807569797
1
3
5:
7.869708833980511883756878552213221419678957105689648113575143283542042343764839498710089836646202256
1
3
6:
9.437930735909170592800736530581260437129272893366239610152950499561279734825940000732123655225447587
1
3
7:
11.00687267915136697198971805565596448438911493160839051360366817406402232592778383923205756672918533
1
3
8:
12.57627081967063730174119625788627194771090286926365651865860175432444725477007508122387967144444933
1
3
9:
14.14597578714862455371445528665344126407819934545771446609094441233876150790893811118316720172520908
1
3
10:
15.71589683548971610162807802326625110962132189922651567859807842954106817746379566009689891305427312
1
4
1:
1.111876398402898292820021098977577839375408619103221062677182479178236459772650724793743196713575903
1
4
2:
2.134230377041531158660604537633043560831670876914871710431827167570541302546625764823755105664226012
1
4
3:
3.169739709838349948160291213428737840049529504725083769752039928014048208750695206996355625200856433
1
4
4:
4.210407335807532364212051449613026949386139024011208745121118875278548368078327310687271449269086010
1
4
5:
5.253487500953784714656685241973134732753931048249103134927819437744921264186189408433188044933339163
1
4
6:
6.297866692112701240633967244785951273361797279439034216149876091653949460228232061089309285628219293
1
4
7:
7.343019342651121233398663798226899325186075632848887336543841800571891104879722524885041887722505680
1
4
8:
8.388667748182989115978503547756896990521864177170568699137716329236789315974845716141504426620587051
1
4
9:
9.434652179850256208104472200723013169070836220891529761995524545551181214350820777201126107744683134
1
4
10:
10.48087454361980382106066499301885174270785448169432738543786381651524178623605383697522067187989724
1
5
1:
0.8471496088852024877260261515341465265759942597803630406709371724482117332436061702678051481552011219
1
5
2:
1.611071650230343502327103494634025792254611535546392758848805394558841947098378035772825214957274385
1
5
3:
2.385316358124829128667828623577996259909956610540170637158850683926461950101371764166144211097001615
1
5
4:
3.164208314440220080401834725386195944150455351839982105205921506594750129007845937705702760969941115
1
5
5:
3.945410679998436779278044266550629961183746473729505698628182614617750855534916295148929835752195107
1
5
6:
4.727898911094895185277583124138610578136450719262455234506270041703212465834954260703379850594282576
1
5
7:
5.511168221477074576911734791364897649898454975929311309980600455724009815473524859148086789918907903
1
5
8:
6.294944718601533924770651801494891457708566764653991734989255214927687168512763785585033481147107989
1
5
9:
7.079068067086339399075653618524265327428276361612942336407403157927325389490175159714519317986764922
1
5
10:
7.863438581029511687106110935414140860257912344106752840025339691584099680483029354090032856576001057
1
10
1:
0.3940941610212603491374322544364538707545678568982816363542352899597918671446257287083507306396497749
1
10
2:
0.7330568659540933841455902074827663598781528273823637343777260892351484750348547524353219702198599513
1
10
3:
1.074837882608933088323949859104884815770084911655697686693153910964294974439441971571468877667934595
1
10
4:
1.418863663564042330413301153389086693828019036951600519360952063760478272754998967795862991184014610
1
10
5:
1.764330515499948787066081425661317124403902190735947846101514615639371507831042996375440013977727211
1
10
6:
2.110730741838070669820648334971640101896986346134813691471269322681536042548033061718332281954648415
1
10
7:
2.457757112716905486532568615201740850638736681555366310869049415113648707887689161269553228015930634
1
10
8:
2.805219102620989554552077828800900336796490252404167008187371595847353436364226653464935763703831053
1
10
9:
3.152994274165589778414444257927319656523648539603226562659080382161110597287324955670745115284563642
1
10
10:
3.501001128641519279026849745172392476582682304285745744371103180374182102839497717855026473513300843
2
5/4
1:
12.68427955763086525568308183251023546545700852319193435604565269835156253414623928802890931288966018
2
5/4
2:
25.19224007073997679559592125746986699989005862713182588524528786490563433039321801690423572817552163
2
5/4
3:
37.73884586804111620501591558211561439673032719708411612260863639458085669843087752287274513240457737
2
5/4
4:
50.29530091122617536572466555998935195525965274624889300686880527672308373748657028771647542951329719
2
5/4
5:
62.85571448305664612809631151859168952156957196464513178793170379742035686053052918347314605662543117
2
5/4
6:
75.41811120660022715800073830402164408713634163723502721612004113670870612237606924944942210234931110
2
5/4
7:
87.98164230327620851136992781848709817292369059757344149785430831431575182150925681253122614428015946
2
5/4
8:
100.5458828014959575708766435410582975085003567888276929381329746400124071780051951268608311020061298
2
5/4
9:
113.1105964118145100663020273129073098104725912902756985324760995839818785923255335358803804829782273
2
5/4
10:
125.6756412848808658705363936047573809614711479851752816348623911850720743012989486423849063996794606
2
3/2
1:
6.474241099242482237657335497393323195214577843357612039126429661763711977118033970786450552411480082
2
3/2
2:
12.66481821202416251826688043115063536723370719126579073870527834101674880096805996894833941404904418
2
3/2
3:
18.91556493515922058317159253066716673996446153753057598553611132361906102226789922978920787589028025
2
3/2
4:
25.18234794361617513918438798424134415080272873087054081631980485946416905801042562698230044379448176
2
3/2
5:
31.45564904737174408735371534103751060479711188674622415094952620495150638576552577572317281092748769
2
3/2
6:
37.73223077623166656964973706171867587699029719781427147620513573759372122757692284291500284780376368
2
3/2
7:
44.01069351897834019144727788787288682541275047402425214721571630701605462416232133402869456602250902
2
3/2
8:
50.29033423041274032003099509304841586342761251542443543333428471217910747051587127056084579317894506
2
3/2
9:
56.57076124748170305113608710776645534448228984994513918756116119213560527336241431747296181304336374
2
3/2
10:
62.85173914977682719859279289523674762146137658268998836276430032494560201378286512064259128763480783
2
2
1:
3.406921426567525345305705110348488170824277492280149861987074769237080006718252946828742363459441858
2
2
2:
6.427765922596060401536868726611412710308044481172257957408568967963988093516673401485721836402442482
2
2
3:
9.522852269953338568840101821385380770747255358627529291140916810868647900373482338871548459037314152
2
2
4:
12.64038116949377966771151062166748408949040969952687625158684119959987875472150679307944779538526787
2
2
5:
15.76734172576594299188861861179078859482707024413466105388155610508444552324659458062485273406471313
2
2
6:
18.89911527413764377130855487983773495623232329297278173072394376556364763303895822469138181117457102
2
2
7:
22.03366807189500377711249742063882956762860905222054393267123881347630958544593849320291532482562670
2
2
8:
25.16996856837492383788422504805585476292327799613444368720043692355810197196108685608982508556293921
2
2
9:
28.30743874721440349763321541317452268576519519832459885979088931188110796742090675299522347891438445
2
2
10:
31.44572986642317715751615950513172714096109793578876416886069305137034588949109195084472262954483969
2
5/2
1:
2.386373474754020981623051609355690058700443240718585462635700992880378309872376522342490131299999471
2
5/2
2:
4.357772675514873396156676719815518327056497803854987746831376187425878427951961685037022689399809966
2
5/2
3:
6.399466982821576207432917882405722002137459462438403139751643023489147166344441432941430450922587123
2
5/2
4:
8.465790286309245149138250653357892516388404676877980693781413867093457181127897747583679434432470088
2
5/2
5:
10.54291911552927622974622372258251958714738465053736309592215138319891912229564271948220013798097693
2
5/2
6:
12.62566156283582873081921093471726357167329613222149666837020322025278206009960222613361986355040050
2
5/2
7:
14.71167538453041403223116083186553142661515739992187648333986087614947917573916771923739510529530864
2
5/2
8:
16.79975735251129144110738084122437115952598236991717021459241399100110746614906368811894967262032878
2
5/2
9:
18.88922814961602026077005645071309906833493178859398453788138537821640353396057485445902767334817849
2
5/2
10:
20.97967591853567812180985191820049022386290196090595755570702662627242157254218722065179958740130324
2
3
1:
1.867996809904417659709920160994333327794379092559240307830528226033273932217751803434209394288025994
2
3
2:
3.323826519463209705779711843700615242836014679265374777304482431233000939202951328770218120143552167
2
3
3:
4.839812448971509622708123391553750780049395186127678061082546824923580822279866760646526069357731473
2
3
4:
6.380429275311487406873592904774880350404309378988548970281453474911818110848562393112173142701122430
2
3
5:
7.932412043419422005867382618223458836950474971961394887041229420797193228706752691468651053121937692
2
3
6:
9.490426940004391798698571761325018065469105884657665552175884759634967737380655368701638664962597708
2
3
7:
11.05199558549444638229561803000428921382305622317425063943735185436109018989342645708885698656620322
2
3
8:
12.61582508411321949338298659916697544992712743789475193034770780335030359468569363252997734211179171
2
3
9:
14.18117886294981950378651383719957198610336660671363420794605284219978495472678035598265908451016457
2
3
10:
15.74760776199527734146433124744222585949278531296659541879873833100698924184967125528124797298437281
2
4
1:
1.329966967542048308871194466829117080872158737071433065260083774717964327710321287700550294003190056
2
4
2:
2.286090254756521868590507106078506119683697462451388411302322735739255128373124353691848629302773382
2
4
3:
3.280374574021406835381473150594837004656805037883552041323632414872493995055067090810425509815059301
2
4
4:
4.296188594680115383290812093870355560573994408719738083168152490890409962368636978953971120149938043
2
4
5:
5.323186389425269957447015025465350906284298178753999727689072371878311592123567208602914359988085789
2
4
6:
6.356439296940952169532995368468862605306573385042534320145250582022812807285533455668386269882686866
2
4
7:
7.393478053498357630728985045479061973339319583977098907732337025301702039128728255258080336053650000
2
4
8:
8.432962998967482677622540689860490934600847969661190455163635386907641261720446816932886326965470334
2
4
9:
9.474113278863426875550693673032447865910613452663126709731303443877255802248604113806910622892982902
2
4
10:
10.51644580037843167008020776015904716964390904333120414408197137489252013135908547247794061775165614
2
5
1:
1.044353654090500974575680921027905489160247656064808178705893830013649038582198807890533114156039724
2
5
2:
1.760788716159371773888036328315845141719042205674139856412906905443510565010741467717766632415664619
2
5
3:
2.498719677323351057889608946548326328366392375410251868724620410054803175900523977222637386131196954
2
5
4:
3.253658809929269876489659294286271417336649673773530164481360659846514711794408670445760665123570419
2
5
5:
4.018702002187411361421949896512337438510282158477382372880030808119987668448449030659046197724342434
2
5
6:
4.789771104525202457203109239182399678396933261333038307969537093286622521411098826536860769435260105
2
5
7:
5.564613517065278403751343253252682505542438507491995700565693391577666540262709180716840125715490447
2
5
8:
6.341942557946664219749387131357496294644418568619983302461848686511942354015191605746430294043777577
2
5
9:
7.120985417399630771478395238188998675978809389320640210086541677563356916197679061652240250849732964
2
5
10:
7.901254887934438255247355119206946959979113347186429679370672477180547962180166161310598751300168958
2
10
1:
0.5142348102980751154013597620621347665330250848760210113836195245816818641647636279410166228779117717
2
10
2:
0.8457405866725575485798446431513513556058299443843549220930230262345512667187593955516687145156181630
2
10
3:
1.173854495726436664566658462637139723861004433425096738838068963494763190028757020321454715715329027
2
10
4:
1.504407579424180025620416900944725708330214814958824765531106090425104429841990799017539022676739532
2
10
5:
1.838338680298082269023869151117965570918977617592844899912613847925055644712112956773575642291315519
2
10
6:
2.175310986472222638045231855490840838959900496424033745406872258870816890351681766065242740432883607
2
10
7:
2.514716960254930258819218998332828532236481415523377533700041936239694413659797334560385062253225618
2
10
8:
2.855995559711866065819065136611994664985866873460869710446702400781666064943666779802779020314736895
2
10
9:
3.198703281647358638469983805987609098949736237017644135799687097674631007632508501602687610459044321
2
10
10:
3.542507388159867854321870669654680121556578137991653154826626465690896725387512781555861939446646708
3
5/4
1:
12.83979263910868002938299029815494184006864244277100073582153197643021668291045820604549044739718681
3
5/4
2:
25.27135610711520376015389909600850405212592826815452438074703786695019483027332824162609789312495707
3
5/4
3:
37.79176046784570828554516623994802705564784315338063509685136615470815363607572683320093904392343827
3
5/4
4:
50.33503178221784993522390471375742387028269729639957710942223180934037472438645979702186752755713253
3
5/4
5:
62.88751583364744309139509055485401949510238116393495782045328341730577335848170661362414426918837931
3
5/4
6:
75.44461987527997360757872540455422656634615991317071239074588693452454061175833525652299770354997201
3
5/4
7:
88.00436791912299368374494010852470299858273036604500527470596982985103926869118687114328984981442238
3
5/4
8:
100.5657699305657873042617058616894155417472110345657143988612807287306100218474893850536870979882914
3
5/4
9:
113.1282752100477771483218370817658048451315723148677031844320776960905634038764577574225731963627471
3
5/4
10:
125.6915530725656024180990659219079482700467851986023263163042166829888019520253296832273982322744074
3
3/2
1:
6.720291811148731588179466372123858304934677022309837761478920412031377890633371769601029861450157388
3
3/2
2:
12.79493818821565115115135356953624078435823780618611972568254721516355552492595935287942341668976021
3
3/2
3:
19.00323581428130879192025109618164017036562078230839619050178617091382214719516734135786389105246755
3
3/2
4:
25.24834600248492164518174787307649737352221186990008086681138109111565417719495786443474138277515939
3
3/2
5:
31.50853848130317211754316483041392048668706418811018976449681301463959943956277176625560583954755001
3
3/2
6:
37.77634655751865001720854652387065457718343036378647648443491477890034978534250670945044128037426999
3
3/2
7:
44.04852838546096282287436447943343373878465412577903192649059656491951240294653366734598350971909702
3
3/2
8:
50.32345186416033812134070223177092445012911370177380587952101505611914965554815496884824151466959385
3
3/2
9:
56.60020653635877589948138393431097157050699841493365710992561877158247737773788074718385185341057936
3
3/2
10:
62.87824466979922677732900250356865182943030989286321591607853059887983844267530816913786366783357770
3
2
1:
3.728870068025545224495134213679699503160591927092235121629264923037786347010493500267660661943124973
3
2
2:
6.615921267944649279842557192751756529628960628442124859461814836452130368310263459584220338879499435
3
2
3:
9.652244999658331673618989903628162152186689997351885224059598998062751621303785367078155490927502922
3
2
4:
12.73848409392549271266712585287044457362338267145484160281986128064561224151644430671484613128068323
3
2
5:
15.84621771295964098851824367603183413512699719519241070755926326238773522948838159816526299771912172
3
2
6:
18.96502373310900199232614026637394146801119482568614288396086315219133212419211293436749035481682160
3
2
7:
22.09025333820531725126662676804972304933282409034848491106224357719791305452127871759127446744058624
3
2
8:
25.21953311377954228591208927416929100573381887378957149308499305155700740664310369685611195304636958
3
2
9:
28.35152808383235403177239061042587885401720198296174755358943112331611999079726657205301617649362898
3
2
10:
31.48543084864751136856862399355912871037954883091214032809841774998510694718491061307311641405998584
3
5/2
1:
2.719856802633922398385419874151283580072784863303198834057867136347739438645868739643205185052581009
3
5/2
2:
4.573883241504167040988015666370467596528547005465184441100022995382998047289593372565366675547715028
3
5/2
3:
6.551841477484810095596688353292143951897728252027214059065905868311754344910820927870436381886259565
3
5/2
4:
8.582306168744259879920040208894756011758374227402232251535102166413284392206451648795545157483354292
3
5/2
5:
10.63695768480849698184135621058995246028427925666388353459136133389392132643977971418566838627797753
3
5/2
6:
12.70439935733883971758379022981244117897506857636665308192511341471590031136672345700441410635996053
3
5/2
7:
14.77935671815747501758374544459148180673808976752445254812035354116766344150190574897524168328410408
3
5/2
8:
16.85908718038576689598090628336185555098638123916699705028027892527010134303561370707145116449362802
3
5/2
9:
18.94203187862597427843006540958837056802879695947164340942062420077141218820962482017205003601740877
3
5/2
10:
21.02724177508066665112619254377716634022508336707727508560218217102675486686764885656859647159249569
3
3
1:
2.188728015921285613808183762636238999642368745643656634646474872273665947505639543972359256092230387
3
3
2:
3.552301341546316255889763860612098438029959120159376348622369195849371919532646302439096968666961493
3
3
3:
5.005627132742489299414140174762980876222835637130015000390488241494088653416667803752624700895279381
3
3
4:
6.508481103456207711088802804395124647694459704843297208268412366626913602864127760389696408336500639
3
3
5:
8.036201749008610191379950957811666727146455395957080883941025529557767685027365569471883279993847104
3
3
6:
9.577519072683806021606300052737196557464512092987168881677073028186931491582065919700465941457711220
3
3
7:
11.12695270574246485123652933066995295189493652131771175761493838968695421298920962139556983294073079
3
3
8:
12.68158523199242726819835854451120147286403275240206954898186010692610808068387296904864453079634268
3
3
9:
14.23973688677902543225734393139395459299025391812100446170871917439835244452009859535271024522064720
3
3
10:
15.80037697883061679682597451742900402696846794754594222045863934390960182621475178111025118587853036
3
4
1:
1.606618177912209404170357835469473433888706638685997466739271994433365632248926475125859537371573746
3
4
2:
2.514895135968527735669075656047517065208063783253037453281736870276593354664966797518694254099457428
3
4
3:
3.457796467433323304147048236619285405471230717001276632098160751033274286953558969779675134950133783
3
4
4:
4.436701528268054007462327665520993508714622310084640379327841943446204318939511321444642853850008740
3
4
5:
5.438292668752278061357346163376073541629575031762395165334573238174582187429602181017092437848880355
3
4
6:
6.453526171553682644114559240306960233064744796100335591030465445713326992340093627257434076241638909
3
4
7:
7.477272860724321754829699561653205434906245664654034525570465672567590600318510816362567005358580626
3
4
8:
8.506600813092704388068055084030531401125986108468121097334679240951441547046625997858781290141011607
3
4
9:
9.539757718203540293867052248998989265128357909231294014532739347385184257233570982989146029658705898
3
4
10:
10.57564490104668235808497905869937867686748942726234174785767976921692645991669500935927573291355266
3
5
1:
1.278919833466644624543417816311204596167016234133979688746611072959940787584822520565997709433234551
3
5
2:
1.974777897790598653630893395974706921872925684192765380134003603490038880863179769238558611636128896
3
5
3:
2.676130210686767800612051985707459526491055420258327166253686480085747018504994264624343364349910727
3
5
4:
3.398793331019829689515063409064435542102426167919744795681438151105947579918955326829757610665599613
3
5
5:
4.139364779396829423343746304502679262658954676482660127917834648014676092548367969866702980219566157
3
5
6:
4.892269028609883389475251342675805194167151448808565032089374663519966888988709512546668459143092620
3
5
7:
5.653405612154483562874362991418666177626616211691610742850513584790876690230001167276322857340379694
3
5
8:
6.420134414613430514817782372206301591421453600513660293671931195386679179647117269503529389951811920
3
5
9:
7.190777316706953759776428317645246155418995886285305628388967639391628678321751225745963084102550849
3
5
10:
7.964245081065915762715173451552346369712740134030767396892611492604359808229254448369456705373079340
3
10
1:
0.6380448251723097323398750245840925614836203290728585742145454524083808971258797583606404215781903675
3
10
2:
0.9764105415008921710235367545317586761902772820889894921596766720341077777618673243001683485794445376
3
10
3:
1.302978932506557804818416514078546783140999881383964330276115394551636609002001208205297611435298576
3
10
4:
1.626813211273345455933419619064379782171241117767943499217170805702197639958019587512821902141751797
3
10
5:
1.951280950330673530207691786706176581753742297659032477968680000642886179584508262311926066204586792
3
10
6:
2.277987746567986501453443652816111646663785657426922739839386230972792198363716744240319373562513504
3
10
7:
2.607536025293105295152500625501376654777984183524421257200559244934698436017049036536355646862599406
3
10
8:
2.939925438942477657179995030716824892416193650715562834171564584854198774219331490398086972637899712
3
10
9:
3.274863169998155390790281407270096041266581893995736829254839700520018413597680070386112745607174161
3
10
10:
3.611964931808773588074266665350818436501947821751760157105270590487458737541662832326427565751942289
4
5/4
1:
13.05437133340393079989991128263170629069089625491893817244105336703418579619506040587330760816013671
4
5/4
2:
25.38170837371437782115934150068696510826409448056270947023700874669374495631806906352306723911535188
4
5/4
3:
37.86571837676912579601552787850494770066624477798969178992626401374754885440992134629829070272700709
4
5/4
4:
50.39060316127113914748621320997574754649468599228379597423994379164548981479927092095607032684096061
4
5/4
5:
62.93201114715913488287380572724586327425455052348053454087268688437859640338683507258716932623353917
4
5/4
6:
75.48171662000679929386714843205810591399863804541303957328484939305887531147617576684085519155046461
4
5/4
7:
88.03617408455724759901446617837746952455842698002280425591580858886622190745354448437205603200049655
4
5/4
8:
100.5936054133743291215333670988697798147441302103601409582790785768707167930837729809390828442695380
4
5/4
9:
113.1530209630208375901451915191836832184572626999311307854245203878786774831565627907870650868602081
4
5/4
10:
125.7138262473066912520947651207171525461267592693564647106988095684950504835120313168516127070452528
4
3/2
1:
7.049901361739559415944669382543474916309436830403682999511819993976842633813605762268302507569926768
4
3/2
2:
12.97497916490586445194829898810093099909164036976578125446480644736496496732455696512320522862007734
4
3/2
3:
19.12532947439615604847187377429352847993749013512812782190906876713786190527847153444049759873027730
4
3/2
4:
25.34046863875556927980250511029677577183930815749311234787938585948526703653780471688473032519739929
4
3/2
5:
31.58244252249866172139277228517761096955038154407406555145899965781955885130351386982224909815004944
4
3/2
6:
37.83802678517719565645047815762030869664908147954680044598388851660811717618869365440853161092767493
4
3/2
7:
44.10144557865020008161513669224764375438281640403960591008616356735797109492869044428207818096254622
4
3/2
8:
50.36978194151445366064720307189138330512780175658221754491685482814968991011718376745839563641375455
4
3/2
9:
56.64140561936920090059078355348036954341798595661997455284157305046536150477431567815744124314609075
4
3/2
10:
62.91533466032538538492977670676771870814017641297834748477237081418844997323058955275670557736601763
4
2
1:
4.133365217680051948950669147579486101016089902840412269592345621136341458188819080183405553288714188
4
2
2:
6.871164033544497312083220723632780190505149199504337904974823240342235712794349923261861366295050302
4
2
3:
9.830862559500329746395457685117724795935601948393455124029612765576041107750046203880100661794864100
4
2
4:
12.87474507907945329115380773699394477334837601214229273554392037479108073721864319532184183253082525
4
2
5:
15.95608605480993043648601588390410950075328863207851227301225457733255227949893732804836464138522191
4
2
6:
19.05697152928440890164127346696919491929962953589519246562453675828883137848059408391376653731875383
4
2
7:
22.16926822279434490249980282402704431915438117909320014880978465231846043935195218328748945432809116
4
2
8:
25.28878630044311418998166508206371397702543730518743908963141013657276936551650174109204319410937360
4
2
9:
28.41315672142274575896301591265357077328288155217208928458872473242108286858172996200182582918726855
4
2
10:
31.54094187650533694595634826410860890436998263957875761071800692723916521054640155790359116930610917
4
5/2
1:
3.115839449643302544606595613164274070693852873218082285327417645497573651675007997162627090325516853
4
5/2
2:
4.860654085804997175061022154844844668623334708284564170069385224997343045939470432118257764531657935
4
5/2
3:
6.760335008362059627209115594723702992888867462769900680837355735947726794510441386738723168169963605
4
5/2
4:
8.743402181425676740108699094672315797124227455349396757357633994948594432729276633438935927917818987
4
5/2
5:
10.76757923750016924607634151082295686359970069342333291787217786578477835073149322403100386601140688
4
5/2
6:
12.81403644906565756078907912866021803919695665128317804545234109591013917424030578523536547933710265
4
5/2
7:
14.87373598337426012781128611392379044375089488370642697435346997864583484415903667164752671331987383
4
5/2
8:
16.94189823665563569907650798162957831908781110980187402987870322641018366872729132223566206773172666
4
5/2
9:
19.01578113760300330280009915816047857531917987573818012008461445951508659175483196429349265905467122
4
5/2
10:
21.09370575547823554547395863256791115189545092400506202624011161679862878484170225373864729971776436
4
3
1:
2.551996414718595606885641369218359124577244847986269683758315477867258867196813169331205838545062346
4
3
2:
3.847615953715645946377448754299312367940665599828378144949028066119415382587610208809409291855616327
4
3
3:
5.230335039449749084351780547701377218059032724217096608551224391265319438793005876155679905555104408
4
3
4:
6.684796148059909680959699218455721500233288613234268589531989287737280416286848483751139155314143003
4
3
5:
8.180055098823814204167266848086165052323631870930297340717913495202482936735807021500552700000225109
4
3
6:
9.698628707466552330487314141516119616477746006093304982106352819416226278549790632868391743069389169
4
3
7:
11.23138470644565729356116683991620406274646867905787874118697957306499158951374214005038557087638629
4
3
8:
12.77331255628818782728418795307903195708348491830439335792953258896946057566859521745076522863547965
4
3
9:
14.32148312402940537802997570675243302771244113142583553772412179130037684463918172282716927377478648
4
3
10:
15.87408339044802331205651894366298180329304668577926997734619998446648249087213929557041136814503114
4
4
1:
1.899601858300470462542208038043539253861066621215549679700177059698962029319554987857015689154855543
4
4
2:
2.792209630572599693886510103819670808966361486881835798305681731877147207660286872538628959109902346
4
4
3:
3.691343967790555512013662974335146869162938980402755933145268202704483297004337092982184852085840352
4
4
4:
4.628093699289547364746236130620425385273499678250905897156088596263246715460302205222923738806557765
4
4
5:
5.597205346593550911831270743252680864984709858006277995148830708608961053319042336732539812112930647
4
4
6:
6.588347108699564354068060757308207361923524307834955367469080906680188724220129255127193807604730867
4
4
7:
7.593970303504930467998031016596587110925025874695428400931405308304157319947972092158920143133845728
4
4
8:
8.609315416434954759246585237585836632704089682555690177907036030665600376336790648932908175634576209
4
4
9:
9.631410193138510661549907938429750770382618672467552776665317979887631618549855304651460319253981656
4
4
10:
10.65834969986240527655994224439139705670245998924765565318019820980356805998033465215928424458809407
4
5
1:
1.518075108009907445963815575731937450432555484878983015390271916327277847805758504100640346420035755
4
5
2:
2.218159922558025162767056232372530787596758232710936392030397084657237880033179247814208542061409275
4
5
3:
2.899358497312843465734299524449348284014344094870447465915261284816023966373957133709072307774268938
4
5
4:
3.592278688571710768613403680110361544179357976278390222149299904532951130355399996245990543949402159
4
5
5:
4.304595495373389451743816842546335370227271185902871710126310037013090790247275833654634331772249259
4
5
6:
5.034283915410600661216210616979684259618870105328519441864525657725319739466961369439334639764892719
4
5
7:
5.777079752968473088469331879984390035338225757935463153661117586132907369501867118972710250803278976
4
5
8:
6.529314774564999454289095467044687776079011388086628254214789966434954251198985465698154134271314894
4
5
9:
7.288349572752089238754993433988326088718080266809635599686148538961842126903260058678971507023006217
4
5
10:
8.052365626586785113628891639425508149735590558590052443260743360538099404054543036195547684855651111
4
10
1:
0.7588352535659301860864067665698611554063693958280961606680781844565664966238591542643785782567216979
4
10
2:
1.106488295367676655659447262095979502664827695451036650336568906016718100820350863588067721987455217
4
10
3:
1.437375047610274673313622463209983306865728469805017898292307013459817131892102690943654520393813461
4
10
4:
1.762121212701645250191093590234536201917070302738325474944579396080221810373237034720248080092524543
4
10
5:
2.084345105142473931665870102025180632119809031569100062472427132754225327106042301224025256544245904
4
10
6:
2.406032165426671221347153858353871520280811541457522103350704470745616173678391949125061725768134087
4
10
7:
2.728562431622460208108953595570664720595232438651035047418049530715052443820943764471626652911475801
4
10
8:
3.052886395533956799614627632951878092620994793582745287334502711163960901884060856038883420003804622
4
10
9:
3.379536799556456500801596359225177713534559979437623378743786432413349700520341891131858087951741265
4
10
10:
3.708683214026376852207946033132841194750478885558284102474264769873765867204363235303663769192361473
5
5/4
1:
13.32513542085475871430539660480759084996969348358977137291134765941407023856091712643433030410394454
5
5/4
2:
25.52289579244846418387966165095054619774183732864199856119865198450584321064164937201809288777324358
5
5/4
3:
37.96059829104836653399940991132702348591922380361245220704436694201639873418791242407565074177346326
5
5/4
4:
50.46196350161866224714529951339876679571413304920472283657729669982297934112126822189351440571799587
5
5/4
5:
62.98917394283855297946949643505242642009127016749314462342477651647276479046515926309953449475210143
5
5/4
6:
75.52938608824244439680455075773980149315166230105583890774373291896941351119430930066521104834578202
5
5/4
7:
88.07705112084163924333513310529258172490810621545622960541956249700411306103976127350149351344384778
5
5/4
8:
100.6293827612793622892909312777914758211160738865205114795718991378082637919273628583425986774327965
5
5/4
9:
113.1848291113144632456634172838433024868155725147593691811404091079141007180348041820960605061594073
5
5/4
10:
125.7424574841195423479838572857333306673601996773564906111827784486152235602669325484012407546049398
5
3/2
1:
7.451567641607687267887045784286993711247050726265202320067405886039260581565273465158338957347751869
5
3/2
2:
13.20296131054449453216061610687779576483649636863716223982718897482192199107509679870604619683377386
5
3/2
3:
19.28122126863970953539355904883464097114492708831701638802827663704262474483193576797005751996939403
5
3/2
4:
25.45844631477106630121990430539060538487539720314378702347010297980262376869773170994559072882109353
5
3/2
5:
31.67722170126119180239200898571659309011552371197237953125339901253645630963497874019325324194893725
5
3/2
6:
37.91719027994600256838377857004284175303184423170023527703522551692340968665007382321145536336051578
5
3/2
7:
44.16939379788147998353633896461175543092682735770883949440844697915426554885586747730205256000190412
5
3/2
8:
50.42929001679253837464832622745262664563914740095617963949595610692839621148061113197925248355851731
5
3/2
9:
56.69433426640320167527228128978704144194226861115516808810681191596127107526500375754131144132203266
5
3/2
10:
62.96299143784387191965185556833276458867432420466012813801547459370407130230687637404173717879565897
5
2
1:
4.595022212481620852426200550042863774477847967371937383083497566351488196068909292998854560184411367
5
2
2:
7.186649727749824729919525899250199944869401755649080593748173986453092573269017898985545538568040694
5
2
3:
10.05639068597778104963315881721732003070886424737383467505274169733807429052982423454933239779283591
5
2
4:
13.04813829444364007695755853546986902974713110451524861159944426560504239483381487438771444444868792
5
2
5:
16.09640848139988033652074671107434182632376372648651638149669486213197423933975955836165441314219041
5
2
6:
19.17464279852883083052614105468217117938410056733885945090831187985386392738159351726414489538293670
5
2
7:
22.27051209211933299621244246253718461073074783031947041814692662584117404025552686780427937409957698
5
2
8:
25.37759295217582804091461228184385953507172510592043116735945618816478466465762952255551686380182515
5
2
9:
28.49222934569711203255957761523872505812638182475767662337927132284449911691355354630221139585650878
5
2
10:
31.61219326762915986971199673313095767931493842084330109282324085197672536289852750015226446703831291
5
5/2
1:
3.545269683058660468464887295363987830150175357786151228309375345006320304882854293916583948978230533
5
5/2
2:
5.205577741739279865636895825272823910755610940091839824551971703137112766046150794224492810438313579
5
5/2
3:
7.020670163750495952296238325542851022152764773930010550553694448650315027359294299229282321762006208
5
5/2
4:
8.947217695210640480736340395281350263033744959998216830338321140632092911951733267806272350365533999
5
5/2
5:
10.93381385546890209977798285553423725136766529026979956238926570804704582235166113644750315673936325
5
5/2
6:
12.95400433757242805027155121879078883236262550327265412547172232957361612481236115311286987068412149
5
5/2
7:
14.99445189135563987062217151944303036875483777055530215797535057899665761453452863402393991906656707
5
5/2
8:
17.04794688315901251496424521705433425507508141789230316723175024124738895532805228168144399217494112
5
5/2
9:
19.11030397805720982130335708782888664421591759830243154725902355422914096647074619006599351280172160
5
5/2
10:
21.17894205070700749865674362630337055308746941225785908724730566109600819104999204251409479130243360
5
3
1:
2.931389205380581483698575952015720357323015819727800225968256448858654570948904728027394755726901269
5
3
2:
4.191101583021122359441757058099048716897255164436641039822696415284905888040636981970404715914044254
5
3
3:
5.507137487432513564766110183719785445016127892547595288574197595690469556095752830061705579814139452
5
3
4:
6.906636869205108308933811896753977561103225998582864051863350650197060322924754944137162344865137370
5
3
5:
8.362624662670833119845102538193927153514157829612263698545389187662201158134866790857321379357310372
5
3
6:
9.852989271964846888147409711439377576989404763141832140538695738549470229951422963352179102353278587
5
3
7:
11.36481159441729444916727715608881878587560321599760410429409861103880815341745638893218092400749884
5
3
8:
12.89068581895056623118915642817949884089908303157991216494843242215499928624673071585700335376290531
5
3
9:
14.42619177912514058504269324049769183162529795030814556333015604217979189244415699239181184276487717
5
3
10:
15.96856216362163879772971881382153324920111567423582987149196884834143193209348746700128274502334477
5
4
1:
2.193360829507835820743440724460087251368775434013528819224670942955950444392593705986040865342901788
5
4
2:
3.092424601025959113881830413661226890249476530352952937664242800602459419004327279085883045476743007
5
4
3:
3.966211617000052360575126569470492530696016824168846739653451933172094644970020040592890971538608949
5
4
4:
4.864347877570070691829896825921056977759347393298397781285766734886265240122612533251696621194213188
5
4
5:
5.797504261345082512264796392776357826314568750133679844367022798092663381104386605648955981041875105
5
4
6:
6.759798100407915043423413984164308361438965300320369514554435025620117949001663372405655560409547607
5
4
7:
7.742987728880457211729081302637917175609928782324479493549679133235166542649380933281279960052850276
5
4
8:
8.740759755575492110544470042683488649369072580583693045494378042640701397166629676357613517086679932
5
4
9:
9.748844584490768098577956424711072063599752619965856919476317305048744442102082301435599597844514737
5
4
10:
10.76440302732474826765042752777383436476683391533356193053001820973712287924900676574446356792224688
5
5
1:
1.754347776511142033046142892431118920951621747960394650745404099624852067077789499470085507690029222
5
5
2:
2.468718828320737295464388612756006041928286376731189618719319712664271768389668526789607221796323291
5
5
3:
3.146887395292288507550775533791948492076063084846457568218287722749824754992521349210925953972137213
5
5
4:
3.821820785032290252587084403641549097670348653308374630693579580217141021308983002756672466925608794
5
5
5:
4.508959367714342791400273005942184430180156956038870962976104795391476027970241365778128152332121242
5
5
6:
5.213637725318127928016705981944263083455585406849990958806529904501794166444288315425949421108287422
5
5
7:
5.934780355925370697567186735148525754072324177719033013204774852859604224617192270104068807291051236
5
5
8:
6.669147052586494945427119567871910884907062306068461033726681063501352492963482526929027231485965282
5
5
9:
7.413572263986690738053098826098139535970707867090673268321991988452753184968740748267473893885761687
5
5
10:
8.165571047779106022661496058421852482231164657705715801804554413858878813105723670617575702104054617
5
10
1:
0.8771484135197071889781914282159710507207164498374409460960774954616346937095531841894369056254535323
5
10
2:
1.233861229590825466122637213233266562604734110304995943737068926597139074980537750038477484486200678
5
10
3:
1.570025270001285519056261261089052456916207246332169133678862702599427838002949477143487117341532589
5
10
4:
1.898058866474819305151669875053525381496037845458597474926561838191991824254789898114479469591910722
5
10
5:
2.221967209850776145708332399785528997085090137826533091649470453511624301668838788242273589118240551
5
10
6:
2.543635462079305890949419688558715873410670889723608621723260026785700501003388902707953539454476433
5
10
7:
2.864253566895916866360791017763038803662041062891808379637016902896213367817845337050320170344199164
5
10
8:
3.184771404575323526533504199001032125134802472226616997499325250902128740256763200775660874946051151
5
10
9:
3.506020122113433321731075722727988634253825987550397092799794115078684732862067959257602027441897475
5
10
10:
3.828686789568774744885885676230078233269792822998826149881807440262130819606250135215769085476061571
Definition
The number $\kappa_{\nu,k}(\lambda)$ is the $k$th positive zero of $J_\nu(x)Y_\nu(\lambda x)-Y_\nu(x)J_\nu(\lambda x)$, ordered by increasing $x$ [1].
Parameters
$\nu$
—   Bessel order ($\nu\geq0$)
$\lambda$
—   outer-to-inner radius ratio ($\lambda\in\mathbb{Q}$, $\lambda>1$)
$k$
—   positive zero index ($k\geq1$)
Formulas
(1)
For $x=\kappa_{\nu,k}(\lambda)$, the radial function $R(r)=J_\nu(x)Y_\nu(xr)-Y_\nu(x)J_\nu(xr)$ satisfies $-(rR')'+\nu^2R/r=x^2rR$ on $1<r<\lambda$, with $R(1)=R(\lambda)=0$. Primes denote derivatives with respect to $r$.
(2)
The Dirichlet eigenvalue on the annulus $a<r<b$ for angular order $\nu\in\mathbb Z_{\geq0}$ and radial index $k$ is $\kappa_{\nu,k}(b/a)^2/a^2$. The full spectrum is the sorted multiset over the angular orders, with their angular multiplicities.
(3)
Put $c=\nu^2-1/4$ and $\alpha=k\pi/(\lambda-1)$. Then $\alpha^2+\min(c,c/\lambda^2)\leq\kappa_{\nu,k}(\lambda)^2\leq\alpha^2+\max(c,c/\lambda^2)$. These bounds follow by applying the variational principle to $u(r)=\sqrt r\,R(r)$, which satisfies $-u''+c\,u/r^2=x^2u$ with zero endpoint values [2].
Comments
(4)
Here $J_\nu$ and $Y_\nu$ are the Bessel functions of the first kind and second kind. The argument $x$ is the radial wavenumber multiplied by the inner radius, so the defining equation uses inner radius $1$.
(5)
The positive zeros are simple [1]. The index $k$ starts at the first positive zero for each fixed order $\nu$ and radius ratio $\lambda$; it does not order the combined annulus spectrum over all angular orders. No zero at the origin is included.
(6)
The cross-product uses the functions themselves. Cross-products involving their derivatives describe different boundary conditions and are different families.
(7)
Integer orders $\nu=0,1,2,\ldots$ give the angular harmonics of a full annulus. Order $0$ has angular multiplicity one; each positive integer order has angular multiplicity two, from cosine and sine. Each radial root is stored once. The defining cross-product also makes sense for every real $\nu\geq0$ and real $\lambda>1$, although the radius-ratio parameter in this table is rational.
(8)
For the annulus $a<r<b$ with Dirichlet boundary conditions, the ratio is $\lambda=b/a$ and the physical radial wavenumber is $\kappa_{\nu,k}(\lambda)/a$. In particular, $\kappa_{0,1}(2)^2$ is the first Dirichlet eigenvalue of the annulus with radii $1$ and $2$.
Programs
(P1)
Python
import mpmath as mp
mp.mp.dps = 130
# First positive zero for order 0 and radius ratio 2.
# The attached generator certifies arbitrary selected root indices.
def cross_product(x):
    return (mp.besselj(0, x)*mp.bessely(0, 2*x)
            - mp.bessely(0, x)*mp.besselj(0, 2*x))
kappa = mp.findroot(cross_product, (mp.mpf('3.1'), mp.mpf('3.2')))
print(mp.nstr(kappa, 100))
Links
Similar tables
Zeros of Bessel functions of the first kind —   zeros of the first-kind function, which determine Dirichlet spectra on disks
Zeros of Bessel functions of the second kind —   zeros of the second-kind function appearing in the cross-product
Local extrema of Bessel functions of the first kind —   zeros of the first-kind derivative, used for Neumann disk spectra
Local extrema of Bessel functions of the second kind —   zeros of the second-kind derivative, which occurs in derivative cross-products
Dirichlet eigenvalues of the classical planar domains —   the annulus entry is the square of the root with order 0, radius ratio 2 and index 1
Neumann eigenvalues of the classical planar domains —   the analogous membrane spectrum with zero normal derivatives instead of zero boundary values
Roots of the frequency equations of the Euler-Bernoulli beam —   dimensionless characteristic roots for a one-dimensional vibration problem
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds the first ten positive zeros for every order $\nu=0,1,2,3,4,5$ and every radius ratio $\lambda\in\{5/4,3/2,2,5/2,3,4,5,10\}$. These low angular harmonics and radial overtones cover simple annuli from thin rings to a small central hole; the ratio $2$ matches the annulus in the Dirichlet spectrum table)
How they were obtained:

All 480 entries are stored to 100 significant digits. SciPy locates candidate zeros, and mpmath recomputes each at 130 and 160 decimal digits, requiring agreement to at least 120 places relative to $\max(1,|x|)$. Arb at 717 bits certifies opposite signs at exact rational endpoints whose separation is $2\cdot10^{-110}$. The smallest measured relative accuracy of the returned balls is 363 bits, exceeding 109 decimal digits.

more

For each order and radius ratio, ten disjoint sign brackets are accompanied by a certified Sturm count of ten radial nodes at the last bracket's upper endpoint. A rational mesh is finer than a proved lower bound on zero spacing, and every sample sign is certified. The Sturm oscillation theorem [2] therefore proves that no earlier root has been missed and fixes every index. The derivation and rational certificates are attached.

The independent checks evaluate both endpoints of every written decimal interval at 896 bits, repeat the node counts with a different mesh, and verify the variational bounds. They also integrate the transformed radial differential equation for every entry without evaluating Bessel functions, starting with an elementary half-integer control. The square of the first root for order $0$ and ratio $2$ agrees with the annulus entry in the Dirichlet spectrum table. These checks are repeated on the values read back from the draft.



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