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Zeros of the irregular Coulomb wave function $G_\ell(\eta,\rho)$
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Numbers
$\ell$
$\eta$
$n$ 
$g_{\ell,\eta,n}$
0
0
1:
1.570796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
equals: Bessel zero $y_{1/2,1}$
0
0
2:
4.712388980384689857693965074919254326295754099062658731462416888461724609429313497942052238013175602
equals: Bessel zero $y_{1/2,2}$
0
0
3:
7.853981633974483096156608458198757210492923498437764552437361480769541015715522496570087063355292670
equals: Bessel zero $y_{1/2,3}$
0
0
4:
10.99557428756427633461925184147826009469009289781287037341230607307735742200173149519812188869740974
equals: Bessel zero $y_{1/2,4}$
0
0
5:
14.13716694115406957308189522475776297888726229718797619438725066538517382828794049382615671403952681
equals: Bessel zero $y_{1/2,5}$
0
0
6:
17.27875959474386281154453860803726586308443169656308201536219525769299023457414949245419153938164387
equals: Bessel zero $y_{1/2,6}$
0
0
7:
20.42035224833365605000718199131676874728160109593818783633713985000080664086035849108222636472376094
equals: Bessel zero $y_{1/2,7}$
0
0
8:
23.56194490192344928846982537459627163147877049531329365731208444230862304714656748971026119006587801
equals: Bessel zero $y_{1/2,8}$
0
0
9:
26.70353755551324252693246875787577451567593989468839947828702903461643945343277648833829601540799508
equals: Bessel zero $y_{1/2,9}$
0
0
10:
29.84513020910303576539511214115527739987310929406350529926197362692425585971898548696633084075011215
equals: Bessel zero $y_{1/2,10}$
0
0
11:
32.98672286269282900385775552443478028407027869343861112023691821923207226600519448559436566609222921
equals: Bessel zero $y_{1/2,11}$
0
0
12:
36.12831551628262224232039890771428316826744809281371694121186281153988867229140348422240049143434628
equals: Bessel zero $y_{1/2,12}$
0
1/2
1:
2.614147111956698209834636250438459760535455138572976960103146790617647388903245290147470986230048550
0
1/2
2:
6.198055727772057440666324802273845554566626406865344303924206612006714052969674581901105456576055789
0
1/2
3:
9.561628122499824933796139123031326564189331300450323173245123077163793990956826997802404931561313608
0
1/2
4:
12.85395100482008193890827001536818209456129787006882982901283533505911615271882956429276818504794382
0
1/2
5:
16.11016524008394511939995092058876042263724947763382481588921380481978648941730245982921378520764753
0
1/2
6:
19.34440440768908500283283760901670023891768167257177547433898815718173351058802463942900613661862807
0
1/2
7:
22.56381649347544694621174396496678486339776005032077906295207805883267230386382243785073209327346049
0
1/2
8:
25.77253202882139144932390254690725763053513149962459017292223870975070693247756234386791689205932766
0
1/2
9:
28.97315845157991017326529863845935390581639242373053831890472422349034866383183086497743116169601310
0
1/2
10:
32.16744924123553030396841896478127221128436713125693673538971274575428984378872074686520724964520031
0
1/2
11:
35.35664118004484050608390053793470035765624378555557037735338612818871328428163809898800056075237220
0
1/2
12:
38.54163974480255690756582363664574712747055280227386870593797510486868513403496357346341258876024086
0
-1/2
1:
0.8842288133994373746785699920506081426687521338609740950808411548430399802221119839563961238806628986
0
-1/2
2:
3.460398879123905369166807283586468974265270335465395627069979412627818672203184738325635839710973035
0
-1/2
3:
6.319967733370467652762403508801247030518429861690598675156068107622177429524176827902652954639950851
0
-1/2
4:
9.276816086323369114938068556569547270452612904651088553125257965764648266491221966336828689262640829
0
-1/2
5:
12.28176414251944386690543834978603358102786778223696979289194336433401187363047989351792508237272208
0
-1/2
6:
15.31518841540012584923459275969739445135463117885660353606999217658306169256987292344850692877385434
0
-1/2
7:
18.36737499823480103222715789411028129848054903234405795402745738284380050757389751652540971881681646
0
-1/2
8:
21.43283125624413677386693918632893770325805848359656714358477209068798712308242450149538160266952612
0
-1/2
9:
24.50815727053590064921877586921164736210286617297652462549299365231186681593796037396752912460343850
0
-1/2
10:
27.59110573763458209177877984634764018273655550427334940734351255198961712923039087285273577617409328
0
-1/2
11:
30.68011534581174048139432491112372716190457795369082003126809670031809657260032531689067614729995355
0
-1/2
12:
33.77405810783845368839688231999658826572676274939674606420535276894241757491582291964507062761623824
0
1
1:
3.774918323528842724353033373133624631292711133710583184648294022438418641695623406827447855517075921
0
1
2:
7.682447336181499044077575165671694904710135322089825236689304660788941567936188225280752347628530302
0
1
3:
11.22256776588613100027123492177512772818250586161848740187592928359936947830301942031363320698328711
0
1
4:
14.64016187174822935398137804692365158208823537441884523113584392107184500187865530039014122200931516
0
1
5:
17.99432080033782497833869438208669718122690500989037526626705989560187288280251718812678690946714698
0
1
6:
21.30928316225529858917758065728325704670653394703939353384448438496126149512984886356351458062050933
0
1
7:
24.59749374771442871296453489570542271969054736639890941743704802411436430181140737101802374693133574
0
1
8:
27.86623010146465219855318652118447998169377359094376023865599130865274783005716703403702644214629175
0
1
9:
31.12013100018910495915946481926873230701638444624164179630847438653512268644923240185090311856962968
0
1
10:
34.36234127152217748033889693345560085026962512803661346566456539283485014148065347800801437427076903
0
1
11:
37.59509430697570035359959876642782885795937723128662207847708399289387486979052643059938371858271185
0
1
12:
40.82003490336447856382185444323952869630322094440534122441670327946572069051376639951932737228040848
0
-1
1:
0.5485529704746079942470022579509325166133705690077306618218524664177867614839236536885044975793497050
0
-1
2:
2.586775191138361763464986974727129440668709029465481968738032012156209032217866086841985911263295986
0
-1
3:
5.129160391352049806021439178059635304147980495360282598309261641207403694436755196713117805460764757
0
-1
4:
7.874434731201421466579503654120802011975613711817114210926515183955704974693209918584183470520312838
0
-1
5:
10.72363097876143554460497789787871679671325596310020360866894797513151716238083739647157228502285386
0
-1
6:
13.63483975909462942943273751154942080889894277351018702347917493598345650038422244349448222340257079
0
-1
7:
16.58690252310300859757912730516820882416979464285088545531571055216739570462365505754689036379944110
0
-1
8:
19.56778020974183648608265147969579727488766170703854716763511159420768513645984213762794988454118063
0
-1
9:
22.57001386017473530694599572820667261671341926639871955615691521234010867785664610986875732211341085
0
-1
10:
25.58868004330866744964289685944226768718135304294007434441580365973336505666088897767437909796371998
0
-1
11:
28.62036640179699724255117587860985904162956667929793384021926835057737956740621986025830939348769166
0
-1
12:
31.66261454536232808481680367132333757331179166184913514584372263627750430982875370933089297562339448
0
2
1:
6.073417930502628705243209699284277869912446194598343761074257064989651301590478859687139349789826691
0
2
2:
10.45957563123158327846813719999423582031512961351558690595080513359476991753067291557564876802870161
0
2
3:
14.28115633202199888545231262170918440568252829213585074914062748048348968495080395731019711187068705
0
2
4:
17.90613014879482266421020081497155188600775154641961841602331705885728996538826507270086268914769537
0
2
5:
21.42644314151869550906229549383914429977181877638245874755747691366140824092387464134465109719599757
0
2
6:
24.88077974440496926984988622218826232427499321917009190129664430625346632646755963532466050104096142
0
2
7:
28.28939122222102014964614366101695512177352425180427829206248739732935129169597719606503756452803280
0
2
8:
31.66430577564667273019578047730151279316415326566361298626536823022823094140080380397965429541787967
0
2
9:
35.01328929490858091616730249110344265918021138154131237427401835652616483479759040836339538980344772
0
2
10:
38.34166395432204769825724357697346400703264335628479518490431165449797205238346415920800397649411850
0
2
11:
41.65324479658362754507874127880362028504178390259490255457032488008328196407943578506576948516415527
0
2
12:
44.95086431656146079128365140983013142028175234450789207038549863784303811098004659488293211664228469
0
-2
1:
0.2943407911187011414571061661928860896690722238878202720257489761653939075291611014424402566611415128
0
-2
2:
1.624029726880959755255047584590279936382589607332509273384344862319568885620865364716394454766741358
0
-2
3:
3.582754488113967193510800080678062286536182357802607337899487243410639292214835328689103975728013866
0
-2
4:
5.891168060552005448408017415576097774209183006380970025072472216488478111554430685889664575314029671
0
-2
5:
8.404716339940085154783198423333508360461785392337493406902889277492595700213611099982200038791197436
0
-2
6:
11.04821788856401596066843023977083715448144243631752017412405679885753029185885051438308782497820569
0
-2
7:
13.77975843080489793510017116621412963820388354867591303165286926211651593756540460916191936593864844
0
-2
8:
16.57422354927378970278265107718374706939164946898334793392293511786376556717147513769274409476882943
0
-2
9:
19.41560440258144645927332455518793741624765804565057926247398381667789373669800945950116100788679015
0
-2
10:
22.29316261917680539872337052682295015348199779483825618269950082543660752592374408877525540904447669
0
-2
11:
25.19938741963050614561133148523629709005552978290428135412707980349133948855890851017267248496767889
0
-2
12:
28.12884069818410136368014186614279249095493772401869978699665761754165223184324873164126745607632645
1
0
1:
2.798386045783887136720248903139570670634608790754101043596421710556249508278535322625506568538426879
equals: Bessel zero $y_{3/2,1}$
1
0
2:
6.121250466898068301287482652303778113635465613353553625062248246904038108457887777868434656613749825
equals: Bessel zero $y_{3/2,2}$
1
0
3:
9.317866461791065379009586644142730386407890410085049660065557215185848553052486482055891458068293429
equals: Bessel zero $y_{3/2,3}$
1
0
4:
12.48645439522378142837734586248318324112703207094977236082137483567007377324631825124089576957559348
equals: Bessel zero $y_{3/2,4}$
1
0
5:
15.64412837033302763004677421391486188035963633197042618687803962692161099188456935690444219713062909
equals: Bessel zero $y_{3/2,5}$
1
0
6:
18.79640436621015716860039941776525240808248774043224927105117775302405893851520026968455189600650432
equals: Bessel zero $y_{3/2,6}$
1
0
7:
21.94561287998104457279185633243848050146065264853951524186146253523180535861498992764558560964807833
equals: Bessel zero $y_{3/2,7}$
1
0
8:
25.09291041211209736004063949486687769553799324580860319684137689412000329633761202276968003697092368
equals: Bessel zero $y_{3/2,8}$
1
0
9:
28.23893657526027292945614678457410549024177931666861464485267586989717128468261864218496794852800622
equals: Bessel zero $y_{3/2,9}$
1
0
10:
31.38407401788985848817423434539121275401424287095219108603086599822523629062587842407536495971620564
equals: Bessel zero $y_{3/2,10}$
1
0
11:
34.52856575546206430777552074790171184370201728938621796239919809998910529638052503662878646667495705
equals: Bessel zero $y_{3/2,11}$
1
0
12:
37.67257356511296615289234056890317665189613760331863685535027339916643599783155335668229932380730336
equals: Bessel zero $y_{3/2,12}$
1
1/2
1:
3.596377387676906236357061406444022023141584893198071019072679468463251040635834892428554483760711338
1
1/2
2:
7.243199747506168115688746459304517381822805426863952909512084283735881939862231738325122423115252102
1
1/2
3:
10.62656869256610624937451958310999392804178517727608825388251123294518002282406294006982039323945671
1
1/2
4:
13.92892686633605477057075848034442426013118679557026814839031466048134154106465625166789827835176961
1
1/2
5:
17.19126232491736485424863909041797344515046920547212984043676633322496391784337229288683869902903211
1
1/2
6:
20.42964074562451392232989577187423925298383710774972891278917004359296299437117150943974519727380941
1
1/2
7:
23.65204457704130774209118702466189877402986174588804103246026330076965487829304948744051583900276968
1
1/2
8:
26.86302611102657332316489859835654365717361166585735736957568049385572713189228765888958048611913336
1
1/2
9:
30.06542964359532526593950933485559890684130646436482150258026760428751628902539624608178199761346822
1
1/2
10:
33.26115220097374209642814777523376644581147568555982795487387684107461536489695023953390792788753810
1
1/2
11:
36.45152274613040604480826948880508236572344093916964383684284886438144049499404627955076918754029790
1
1/2
12:
39.63750871308825524145984256450797237710309605252670643811101426601756012170044648859628776122519641
1
-1/2
1:
2.157872407894999253442678748790948980938936099228346025212246291991602772793563720954554394638241670
1
-1/2
2:
5.127274099925793749114145463565146046006886783261536299543877052934196792466545364043417068757136507
1
-1/2
3:
8.115201665540024400725417857004579931806894608857882150465414933782227749461388983298228206915920541
1
-1/2
4:
11.13479614874432494388944174917849453112985808739950280225328484541246554691827568510731873032043711
1
-1/2
5:
14.17671258016775987720164790563231230877376601939219070321181461842679026130771965832629369901758539
1
-1/2
6:
17.23443083634809597741746092835397906996308527914003326472363816011344136560759271915949839442136546
1
-1/2
7:
20.30377149064861067700347025840316944442522429954678443625103277888411852247474103676810801091168281
1
-1/2
8:
23.38197260919567052864405981728073598691561109607294905490602640721911703131621208307765199168382221
1
-1/2
9:
26.46713372719790873912774664162597149668535188082417063365652211327082113486721290744916132341949136
1
-1/2
10:
29.55789809849865462570779308334216244778540998080612204029042138531301412074999903614270691248385329
1
-1/2
11:
32.65326610089745216636102217134697940426179192531895849566156597429028116938392382236714590341559137
1
-1/2
12:
35.75248127950149816861418213488630412000122701826045606558773256581254157580229760504594289193375976
1
1
1:
4.512252624480407772267928841819054227132367691384909544458067766039208810101342324710345685805291608
1
1
2:
8.442260136534789697625868570853756395674908185906220263532448301877797057429952576020764965289371046
1
1
3:
11.98998196483788624595844819104080303507250528319269088123700881237344192326524913706260340486590859
1
1
4:
15.41158243449326655730261357909115085773408833755527558763724486156671580061790984023360583740845885
1
1
5:
18.76824852843797979158898132091951219781501201899224334563973616286657067786365039231616220008549160
1
1
6:
22.08493788487968326152685209631802175010547317331207851921203415359313732723692646121239128642435239
1
1
7:
25.37441442743900010973606996716419069015885277016314043628278439309699640690064283270853599763202001
1
1
8:
28.64412043220898336965775931161596092879475325712759847179006154577048236503927698684207155860920399
1
1
9:
31.89878876422869794771762904100267206492487765099238768760732075219462524440713205074927926245153356
1
1
10:
35.14162207308293652989493874852692499029459645502745854642598078476849342264848863124654588138772433
1
1
11:
38.37489131993098662624014907717057838194444940315897472161139761307037040658295700276432405706854865
1
1
12:
41.60026680831792448418394086616610714902382123190240821581008938661908626266127048147306070803128323
1
-1
1:
1.680964643259288730819696698419297814351961992420592430985084484564152897702087738190987788750666180
1
-1
2:
4.294690026186120877392284700915573586561731643947682928856459332942979113470368082251013171424248570
1
-1
3:
7.058943952805816331826104755273576563021597276147552015748219830728439732207674189597665398365062785
1
-1
4:
9.916743466380088735129867034091155751888347575760087889512098376529040510297388032551939581871225377
1
-1
5:
12.83280700459182127207402257249768884997175032958646482050008112602178187800589288891416266650344743
1
-1
6:
15.78796917424384745640925307896160880490569567625743566642456676136306010897680619356882154414562495
1
-1
7:
18.77098974462645010924739318963712167602764667675862716778450704262329911673643302446889630881059853
1
-1
8:
21.77478993906867759563808546188474363816999027139789639464681210052814423534527845292456586917914201
1
-1
9:
24.79464954100864966639136181631062893218052751798285098870216168220070428484211340721867713384764590
1
-1
10:
27.82727435999770539716250451115903450698276850207388865033266223231659565429052037888817778029374242
1
-1
11:
30.87027926493355616513511580763554755222709915622679797809781694777267334073646619900565113577336043
1
-1
12:
33.92188438471240577118907653366088512934891638743624359982108353321231938546010203876851614883306534
1
2
1:
6.525856323522473886312399993787942859349765048887598725545972550562678218480448149574260032030355099
1
2
2:
10.91653799033526222914110652546187322290399663108684765592537547022563931465989755657338005078475228
1
2
3:
14.73985742124029528792479482497814009293461687440399571466604791558183518010024544089106429726511095
1
2
4:
18.36580998134117863132684699129320138069136870143148785492818912226464820753849845058684072706286729
1
2
5:
21.88676247178675246848310977725402308435781074878059244310795783251286990102223476054607988541363217
1
2
6:
25.34155355640753657056271886436035240817821119491962276837881755359103059616141756979261515900721178
1
2
7:
28.75050627392511944879885985372470921047627207109103818629215902526240033549916055786707884205611859
1
2
8:
32.12568724837723980857426145139887201182145920487584575227672311405153725478833146860148733142722017
1
2
9:
35.47488496737540093972338239498287960528095730875172028661395462623188419166722888918300495762269041
1
2
10:
38.80343583226864515471806287932675998935708554555680670936664369034223816470160442113142601955236291
1
2
11:
42.11516432040300670217291306234200547319438121364198825808037129524457887515010788912534535632255561
1
2
12:
45.41290944531147872688965261801538637513477712662827698201474656136558105835068026274678162016186329
1
-2
1:
1.097747630517234691451799430471236435397163436592956156476984078519525582322844296875819863464933131
1
-2
2:
3.096000870899929719519525024549524364050868800873252997966325130568526208734460264915331919787219674
1
-2
3:
5.414208671392083958823357755716078245182890226828616123622672787282250192780092377256824267884907674
1
-2
4:
7.931955692527394278832210506232159896097232257644105734174530996841118179157968533027828608540878750
1
-2
5:
10.57772628963853143716153593334073312447744839482755249655507903723836333676655521731725570133922306
1
-2
6:
13.31066707641854882788080465228588034459874649651401007110415313501072524698788520339976454411733874
1
-2
7:
16.10607430885410506498113904942836937325627066907608968708846255134598345681625102577072606305784470
1
-2
8:
18.94812873594663743053772991199039348553776088296674028055155850048402530541323240820367334504829736
1
-2
9:
21.82619068004841861826295225931916805060491684812067126543209754885864449127177126781929858503786247
1
-2
10:
24.73280544446002803203705875496447925362062565243724034764559284322246464643999157209704263817548494
1
-2
11:
27.66256899893870345034868141749033426764571671618567187164963608874712797321612559778217107118169515
1
-2
12:
30.61145038690465982680929492012907464172622315272453731428391500338430982215016315542998381458536108
2
0
1:
3.959527916501095353234787878643086969363961591074408467623667640707991794296505256906680644869728985
equals: Bessel zero $y_{5/2,1}$
2
0
2:
7.451610064214503401566550454738233339402897542846858122368297821854134665950908072481481540087249191
equals: Bessel zero $y_{5/2,2}$
2
0
3:
10.71564737579151256686307814095084374364486037156945098871575577663458313613505429579462575258837420
equals: Bessel zero $y_{5/2,3}$
2
0
4:
13.92168601230878169402864948308043239836748041751876989012735311131206771025390362469822120274339600
equals: Bessel zero $y_{5/2,4}$
2
0
5:
17.10335911720874089901860000679832821187129970418103457757811061074375762338783205222584093695311964
equals: Bessel zero $y_{5/2,5}$
2
0
6:
20.27236914021652938308320744435360931363063417366375904626086502893244700603153263955675079479117158
equals: Bessel zero $y_{5/2,6}$
2
0
7:
23.43392614206780242975428657779154226438478549011227350655478421772564334885903057933212274480152543
equals: Bessel zero $y_{5/2,7}$
2
0
8:
26.59071663108627108543275382411055013051760068087197099373861964307042246032451510354863080904544417
equals: Bessel zero $y_{5/2,8}$
2
0
9:
29.74427068055655639674584086447468181654328807428043464368770616600716406224106052659519424459450172
equals: Bessel zero $y_{5/2,9}$
2
0
10:
32.89552518822430415043798901378285704530593133814875692869712915916397858470202215678430606191708039
equals: Bessel zero $y_{5/2,10}$
2
0
11:
36.04508650792244526382337353894329059514400065680601362385602765693012460963139828310538308188711262
equals: Bessel zero $y_{5/2,11}$
2
0
12:
39.19336465949011217165339291343306913016518231748013735924753692411995025000752001474640962002717162
equals: Bessel zero $y_{5/2,12}$
2
1/2
1:
4.671234594934786984102457660600780969320584231901133850420774758946391025991336451197958928901898442
2
1/2
2:
8.421150080938223667621971121543301567827258153085590859302556534177297331332538366551949991881039639
2
1/2
3:
11.84501042315228633033084756503793867323234875925849295809043451620271434487836734169337386216203765
2
1/2
4:
15.16995525993688886304193453760819941020446418234139928663193223101943782829375716967316839575698886
2
1/2
5:
18.44685995774562504449637613539200038891825857047634067390494938844440931346759928915869022238234314
2
1/2
6:
21.69546288427709814667574169146247305436203724191890230441853269057864045323521920595970222308637240
2
1/2
7:
24.92545581899146463823439649361048330304013173175466108532265651101008585459449291758129194647817643
2
1/2
8:
28.14230176526235090273879714046272492508506890284125819589902869008109595742783277470456836463850097
2
1/2
9:
31.34937704765602618737218545502905027498558833530012561486038133333948016857146787310826438816357820
2
1/2
10:
34.54891118999203363025068979581010856413584606653326304652716589404836990844018469420093128409022550
2
1/2
11:
37.74245200783093421509804979560811317877258178559405991416069553392162459674892549584369510904936971
2
1/2
12:
40.93111704474807854525509381702071456614286612678167866569524536072973045043964987786567335517827796
2
-1/2
1:
3.346589635954815125192504823281829054032310713797036179227388505347586669550198426078061718064272499
2
-1/2
2:
6.569347656904348524011282203466388682502553858757137581617803510986738171724646638057895257444795278
2
-1/2
3:
9.662749710799260330673525982718283285205703413958706143730118257140323008973963201651368065652967213
2
-1/2
4:
12.74138253753412853332153864061820014511385294530944740608296710200881487399159386909641323691733407
2
-1/2
5:
15.82111570945920981320259988102030613315556412072373805794785548399566723553843688402977100532039505
2
-1/2
6:
18.90512888531925936631174293880610617996629014439479538820457216756045101716519952310374370887666304
2
-1/2
7:
21.99380617950025582629176135501523429156824993043498010275342921300979999818396300978386144638797225
2
-1/2
8:
25.08682013205547925784980777983698648207959616322259512858121596445588166059190995451937825121128002
2
-1/2
9:
28.18368831044217799508672679193054258758376426527619812497417648149338119612120077077700989814056611
2
-1/2
10:
31.28393592255738548076589976409610367633091007195349100022685692980185665034631507074932564611681673
2
-1/2
11:
34.38714060071679089046283607988693991884293830409001301660954846314013122079218385838661979464256910
2
-1/2
12:
37.49293950336645565032667394985334111778640646029305804786363039234241418518983528516101889994138109
2
1
1:
5.469296978410142274283885874071685969950310463532589347560517127840388158101620267961385872810037920
2
1
2:
9.459379718201374236620260213764527760183127051914793338398918835590369295191584594081487121650364101
2
1
3:
13.03089820742391542348063810960213249298084386848425810402506829685704965446091010314687025297622279
2
1
4:
16.46593377865154183352948129531027102756135314037678971115287241773033957001908386829960212960703585
2
1
5:
19.83135641408479601800077669859095819727201679123959892736723970170026644322647144926165299402697592
2
1
6:
23.15424426478351758203441600329744374534119982528651812668079237069655093087094151317378383052248864
2
1
7:
26.44835493843768289441456198488053349985201892233029682948561724817861572739440234656006004428259088
2
1
8:
29.72166386587788611696462605791116005578495567925744719997791619079316311297808347072278194995378437
2
1
9:
32.97921744998980800042891889384386851176832299924780288251685492069199437730866619516736666933644769
2
1
10:
36.22441548893407886080424582328429342990489104232980575830528197305916142617661207908699557224974856
2
1
11:
39.45965943191465052364530533522103269884368813141852346008005574965383175434656739210865075330076240
2
1
12:
42.68670953670101987489754319766349187548685909088356579281332658769961525023895691347295804552422149
2
-1
1:
2.835425849685837176253259549547500508828935685862524712306790001527934093389590457600517532522543363
2
-1
2:
5.787032834006697163872845719438112239903674615489386416359044524400929921362152408750787172986804280
2
-1
3:
8.702199645760309191556279681570236280508787737848220238648788997766219374450404061058319343470529785
2
-1
4:
11.64627846830258220727598670887734012370599437217533312656819201457759138861034657447316188773112225
2
-1
5:
14.61800419952297258761653007781942446216026247074761791395174437209371227477431323557149023387008293
2
-1
6:
17.61194246166105514662571096726171605888174598815034565070707226918642198221507413793975094086210361
2
-1
7:
20.62346815707151130054779533026624422257093390411720674847597639454667327697218199318495953831165044
2
-1
8:
23.64907573077222656654185829352760070019440024310605430198254894925979779775871153945494709277935079
2
-1
9:
26.68614029021251509441622451593896002483891299096419731315784531980937499680510251981907458685734667
2
-1
10:
29.73267541564996469636373659342009291218500131115148395473800236710225974193799584031030550406445974
2
-1
11:
32.78715333496517892109188398668638350923770109647242877373301679984778066291844322355516238595322589
2
-1
12:
35.84837908135261663297223587984791318869706507745398732063696416318618424459080111393672121909917089
2
2
1:
7.254342683667989515107608693811763943371135944049041532096130968526053765170534641564468767867804550
2
2
2:
11.66600694024955647579736298859343588635171177171522715856843233224373327247808681929288322816476006
2
2
3:
15.49804735009411300889743337181324668061807757438961321861736362329062532999526587419895577896234315
2
2
4:
19.12908881956323362404150585243665290863630107546567587436599040906151074261390453341797050399262042
2
2
5:
22.65344032119810406838908867780389198568379196731703079266980045849971928123178670135743865411894478
2
2
6:
26.11068386083486750573525374590144966821076971069983776819958167581865899205327085299940657003238816
2
2
7:
29.52149858088562874103333779519157158244920893632199145321796805269098984683535248528583470157139795
2
2
8:
32.89814582848390430249694236463170940144024823932487111641309705313006151820469764576219779413830067
2
2
9:
36.24853049705126629326205049411994864906584311142756640316529841528014606601087235967138434006446237
2
2
10:
39.57806323655765657882041886595472611303691493045400120951624336794765559574883579792950851188893180
2
2
11:
42.89061829004407666297018522408319607895390700627087129080137467017501678490373003368817924324499588
2
2
12:
46.18906936307522867502898535118973792964429115505737016166279794853152452575572413060082781521576527
2
-2
1:
2.084581917255670461050368097938842794666849751617345923358860173191372059301909589466765595488999963
2
-2
2:
4.528936351036075600422329462591227526289277128758422115140232958176923814319432205042486140717498469
2
-2
3:
7.083799480864232918418004838856630915029704805672439778831993115702335971243143938480165726815033445
2
-2
4:
9.747130197983298639629815784131986561515196524164636549744645069348005815350320354710392409779941320
2
-2
5:
12.49019050130812559882580372667783479568277493611429849625932761261290281574528817737906828226896968
2
-2
6:
15.29212949377385926884446112426586051239446474733388671933344023455656422303005815199588202406074508
2
-2
7:
18.13872609359434781367622965538968347565914761123589556780516493401267255223603425356400071293637934
2
-2
8:
21.02011832335054384118430639712324560179714085783165493253766908704984513895343795289606339639095829
2
-2
9:
23.92927340515517165848121926525454273660002027473572134557811549739165454226014890280753402017233360
2
-2
10:
26.86103515555664669108509185045331447369329421891440042606342354496904142138642396298423581450005511
2
-2
11:
29.81152736050450838689724795586531955115604226939962684014680923674229659627655954176578458062009924
2
-2
12:
32.77777105353832536488676613377388612531447715206164061996921725070998163151758530732082095357687675
3
0
1:
5.088498013940855020460112459999430625439597267825676228757503683726215904987369532866090255088267886
equals: Bessel zero $y_{7/2,1}$
3
0
2:
8.733710322592905851375055907451221596784320810514047700715699312781892140095374054934197196664399273
equals: Bessel zero $y_{7/2,2}$
3
0
3:
12.06754368609804606669778359119619804936134085695730661797890205057505775925779780825502415824099550
equals: Bessel zero $y_{7/2,3}$
3
0
4:
15.31538968115123808528162964764258666997462222795706528158184079964128185168330905618339651454074583
equals: Bessel zero $y_{7/2,4}$
3
0
5:
18.52521037214509329582433060571105360912515970362757153617401953683903471522928183219722561096337453
equals: Bessel zero $y_{7/2,5}$
3
0
6:
21.71454728634872122334913273560791254071660367154148764082645517217163623938408669953750419679199277
equals: Bessel zero $y_{7/2,6}$
3
0
7:
24.89150269273583846720756073919322797000879685185861806015035050702569996736051260759760105250662486
equals: Bessel zero $y_{7/2,7}$
3
0
8:
28.06037459290817946299479233507936404727896145236081870462573693325095007656009597325566537407427615
equals: Bessel zero $y_{7/2,8}$
3
0
9:
31.22366672777047819802042100222135607220197902077970988523817402136265064943212353021062688269450921
equals: Bessel zero $y_{7/2,9}$
3
0
10:
34.38294065464467186456202018751980541292536466994840411010127979680589432538970986703989803324561366
equals: Bessel zero $y_{7/2,10}$
3
0
11:
37.53922257429682146323738829991732125263658630584347378050334798284762215231705155776272900209299482
equals: Bessel zero $y_{7/2,11}$
3
0
12:
40.69321542897016522316893178525159359705533080442539879863905988825683692837167617478474117895700337
equals: Bessel zero $y_{7/2,12}$
3
1/2
1:
5.755875112973920702877064043460752761439329398371643274672957283715775833807681361590104891141684336
3
1/2
2:
9.617660705317553265262300373583838731448289852799106414454502171080558143113050341808214295249997168
3
1/2
3:
13.09160206562952027233787008148371203756989751048173046751784857864930617779162987343796442932386045
3
1/2
4:
16.44646178941515965582263221202922402332936157894129871490700871718428879449469814136346538385043642
3
1/2
5:
19.74352752219819992091097004948238567710315708814618454859786434175673310458645623710972540790853264
3
1/2
6:
23.00672208380249341051116848916659814686899623837004274085245279008296540854199931796582470087659046
3
1/2
7:
26.24779715160315414545044441326276729109158252115801006164266756122332114303041783226068925805080977
3
1/2
8:
29.47336060055753536462487815863321679177535727412954720404175971858647704842620622223997914387262055
3
1/2
9:
32.68748007202503251209125061573205449025278631266298283321804074806415083351803176127302944291506662
3
1/2
10:
35.89282879903817220441793065333187735827831545032154304383814070160439338244502663891720902218482010
3
1/2
11:
39.09125312551210755654341409039364668349277547611059761711947103999702911282877781882577958421509418
3
1/2
12:
42.28407915235397163895661934414651629541205381449178313575467788038584028439208241070832073114152451
3
-1/2
1:
4.492661970237464052228001453347016903329516363664071490135250756209446325263117751558684395655470071
3
-1/2
2:
7.915696484818413021272857754897974355452084351870552593480718728662323584958720018808937579939271049
3
-1/2
3:
11.10320318979621625259366472565975061473051513371503296129101903060240101841748754600025605634609163
3
-1/2
4:
14.23870502802030841036829804458209962531504123717595847835779958374082793212238010395074889284623988
3
-1/2
5:
17.35682243785858679987644832872156221106156757468175057661821188135297747020976567046415622883005613
3
-1/2
6:
20.46855576219519092648970322763103736773830556065441712301080474345845930769982964205416295622913903
3
-1/2
7:
23.57821042097536719562064908973274467037410201274711342185450695213778684227270332153122058282677168
3
-1/2
8:
26.68765838754233047448215926117599652990843974227247585357506566822485815890709041369686278949449630
3
-1/2
9:
29.79775087390468777483358587437153666533736456363084446785946034850479275430174044983897549745085193
3
-1/2
10:
32.90887020596523217165789454269174525014783365117333606545362013020983854277016215978817415678866218
3
-1/2
11:
36.02117049404091169292420427781170940329063905957261948660265965877905491006575490400667308989962966
3
-1/2
12:
39.13469109826723040336711450972011572818715227174816470525024593905101003631847568709534038633377758
3
1
1:
6.489408197720460070852422214336756292162826241822963431664800083996683707878788268680822081542995036
3
1
2:
10.55867654185356561181852520454549586236097113659567162746590615048495852985637469019437728129110901
3
1
3:
14.16541344323707961070315166261456006225056014169343685348062895781570560364310926205342408461458315
3
1
4:
17.62153278676569963973311012005967325262893438756959108928808088190035144424693929173060910595061305
3
1
5:
21.00121842631741723913613782479227098447488152418914930101375958979553740029419785916791297151488318
3
1
6:
24.33446983644721487369954772522200734454158068641037459806574260629943723957864679629690804993484888
3
1
7:
27.63648079122989970998467373838485780182846246154684617356518688054498610072013281465658074625872193
3
1
8:
30.91602551078171875918249768667550027796176362155358005482795932952768172330210323357250605384635904
3
1
9:
34.17863355068253371002889716257487002498324996328858294565614494101719839408806939753401116042732984
3
1
10:
37.42801547084071323802379742267664687010409682894216158317838846267017630232579559004957692533516147
3
1
11:
40.66678228277066166704858639097543094249431036347967247346390665743433204749181741892054419449964844
3
1
12:
43.89684099112648366274211715357873531076351303263444196647117885201707640619064127073010756611404731
3
-1
1:
3.969971312628938729950229494340522022280268560549760546410397633529476573433201432726005199880490585
3
-1
2:
7.169829799624000526125877288114613131800271468117552514209446248723913014905758620229149209150726790
3
-1
3:
10.20659757773850652182385888238449324202374201715637801713444297845585943498064829990502439191761849
3
-1
4:
13.22529135861657274537943514964962772796563615519636538572663735829687442349179053045669205924702092
3
-1
5:
16.24770534769891418104222391269673817469818178621314261748525675527673744102796047007796650124820222
3
-1
6:
19.27834611467953137882259211228171075337549099833399130935510958075722659290711149339289116417357201
3
-1
7:
22.31766841165615256487071266502808451405180864988184559181496581519893495405787071459361715595254362
3
-1
8:
25.36504859094623459415592709205351585831691757413450643015435849682992536709203026148471822336073263
3
-1
9:
28.41962155903131671441093720353645965861037837244001298072540600015216892481421545510049262007022038
3
-1
10:
31.48053686028444798345362993708976078102934670967250779228260821356672513781017551497863795704964580
3
-1
11:
34.54703280789565991613300084256257531681110336141321118970769016658556082723071819971259769452418098
3
-1
12:
37.61844977636417231443921349110447759391412403444015697398893727880848865649615581455984398507672385
3
2
1:
8.121481385129127782159559942048485163847392131626314083602929402638562724185119183233831771023415842
3
2
2:
12.57149801326895492455990841118804619979432740749902858165546642293590335586104556306332222621207656
3
2
3:
16.42078777061783969178733691156176999181799110040210498369441955499612629333896502268126102873164184
3
2
4:
20.06230119966950234387803723256630777465172449791053298661115727835576645960492589300973269625370562
3
2
5:
23.59382631370393056301569840811604525955404451374240158841356388284893962656123924601775232964525701
3
2
6:
27.05633907650851632585255316566052527166173972454344080466505290554218951136848269725674583201076092
3
2
7:
30.47120840769154338387001355640970085328641864659234159769653342449435188162568051090303108656024741
3
2
8:
33.85108218922492737987034658097949857583573561796186718986210972614488185000361567678510187206635640
3
2
9:
37.20410088842899132192376808872809473285754348061159062491401532953597496511957656373539557312677238
3
2
10:
40.53582778507089756419902875766438209529100908027493040571058148699919819325045092504361042137547988
3
2
11:
43.85024076022901735509070137538052966074742477972140032962921400587405050358687879777238327641718287
3
2
12:
47.15028656233052376647847219876331200650188034986314473562305868044449410834397053868149213193996644
3
-2
1:
3.131381082733858367492512158562430624175010677596411208279474240353359745562019213786206122965799668
3
-2
2:
5.901549586100540764056434859545068623096409688770494649396958133115086958547575821492320711558270854
3
-2
3:
8.633298706795535461109193017181807350755857303942878221264828003407087643464626586967322393881276783
3
-2
4:
11.41035376369445839518080648467817008354667825343466421647678819146911365412759316505973227415079400
3
-2
5:
14.23256805107020329422538529667597124506627047751744241814580202720186153194498513393792497023219182
3
-2
6:
17.09258809561429744267116315230651959719158537359450475998012595446042576513148214372421274728760137
3
-2
7:
19.98349849409100404245371993337248660336178735551037123431145653240634178009368559925813861799122895
3
-2
8:
22.89974031216792508590832071997762646321765776094299439848845501933580629644311836210594597328097450
3
-2
9:
25.83697532393799195367993094282115785521549932166285828706360706045945909616695257515689286590501324
3
-2
10:
28.79181702312302933025009811659794021684450432490358420845025194082380523370565630639516212621848682
3
-2
11:
31.76159715065330539351717867142813381405534503608667915249055133192465636427201313612362977803220606
3
-2
12:
34.74418809052977820152110721377828900793454055982249864964113842981260506353831110475024184160376854
Definition
The number $g_{\ell,\eta,n}$ is the $n$th strictly positive zero, in increasing order of the dimensionless radius $\rho$, of the irregular Coulomb wave function $G_\ell(\eta,\rho)$ in the DLMF normalization, with $\eta$ held fixed [1]. The origin is excluded.
Parameters
$\ell$
—   angular momentum ($\ell\in\mathbb{Z}_{\geq0}$)
$\eta$
—   Coulomb parameter ($\eta\in\mathbb{Q}$)
$n$
—   positive zero index ($n\geq1$)
Formulas
(1)
$\frac{d^2G_\ell}{d\rho^2}+\left(1-\frac{2\eta}{\rho}-\frac{\ell(\ell+1)}{\rho^2}\right)G_\ell=0$ [1].
(2)
$\theta_\ell(\eta,\rho)=\rho-\eta\log(2\rho)-\ell\pi/2+\sigma_\ell(\eta)$, where $\sigma_\ell(\eta)=\operatorname{Im}\log\Gamma(\ell+1+i\eta)$ is continuous in real $\eta$ and is zero at $\eta=0$ [1].
(3)
$g_{\ell,0,n}=y_{\ell+1/2,n}$, where $y_{\nu,n}$ is the $n$th positive zero of the Bessel function $Y_\nu$. In particular, $g_{0,0,n}=(n-1/2)\pi$ [3].
(4)
$g_{\ell,\eta,n}<f_{\ell,\eta,n}<g_{\ell,\eta,n+1}$, where $f_{\ell,\eta,n}$ is the $n$th positive zero of the regular Coulomb wave function. The relation follows from $G_\ell F_\ell^\prime-F_\ell G_\ell^\prime=1$ and the positive small-radius normalizations [1] [3].
Comments
(5)
The irregular Coulomb wave function uses the DLMF normalization with $H_\ell^\pm=G_\ell\pm iF_\ell$, where $F_\ell$ is the regular Coulomb wave function [1]. Equivalently, $G_\ell(\eta,\rho)=\cos\theta_\ell(\eta,\rho)+o(1)$ as $\rho\to+\infty$, with the phase in (2) [2]. This fixes both the sign and the possible addition of a multiple of $F_\ell$.
(6)
Positive $\eta$ describes repulsion and negative $\eta$ attraction. In scattering, $\rho=kr$, where $k$ is the wave number and $r$ the separation [6]. These zeros are radial nodes at fixed $\eta$, rather than energy levels at fixed physical charge, because changing the energy also changes $\eta$.
(7)
At $\eta=0$, the zeros coincide with the positive zeros of Bessel functions of the second kind of order $\ell+1/2$ [3].
(8)
It retains all existing radial zeros and their annotations. The s, p, d and f partial waves $\ell=0,1,2,3$ and the first twelve positive zeros are practical depths of discrete families. The free case $\eta=0$ is intrinsic; $\eta=\pm1/2,\pm1,\pm2$ form a bounded exploratory reference of attractive and repulsive couplings at half, unit and double strength in the DLMF normalization. These magnitudes are noncanonical references, not measured constants or a claim of application-specific demand for each root. Fixed-coupling radial zeros are not the energy spectrum of a fixed physical Coulomb Hamiltonian. [8] [9] [10]
Programs
(P1)
Python
import mpmath as mp

# Work at 150 decimal digits and print 100.
mp.mp.dps = 150
ell, eta = 0, 1
sigma = mp.im(mp.loggamma(ell + 1 + 1j*eta))
phase = mp.exp(mp.pi*eta/2 + 1j*(sigma - ell*mp.pi/2))

def G(rho):
    # Real part of the outgoing wave, DLMF 33.2.7 and 33.2.11.
    w = mp.whitw(-1j*eta, ell + mp.mpf('0.5'), -2j*rho)
    return mp.re(phase*w)

# For ell=0 and eta=1, the first positive zero lies in (3, 4).
# Numerical approximation; generate.py certifies the table.
root = mp.findroot(G, (3, 4))
print(mp.nstr(root, 100))
Links
[8]
DLMF 33.2
[9]
DLMF 33.5
[10]
DLMF 33.22
Similar tables
Values of the irregular Coulomb wave function —   the values of $G_\ell$ whose positive radial zeros are listed here
Zeros of the regular Coulomb wave function —   the companion solution has the interlacing radial zeros in (4)
Values of the regular Coulomb wave function —   the companion solution $F_\ell$ fixes the normalization and the Wronskian
Zeros of Bessel functions of the second kind —   orders $\ell+1/2$ give the free case $\eta=0$
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it retains all existing radial zeros and their annotations. The s, p, d and f partial waves $\ell=0,1,2,3$ and the first twelve positive zeros are practical depths of discrete families. The free case $\eta=0$ is intrinsic; $\eta=\pm1/2,\pm1,\pm2$ form a bounded exploratory reference of attractive and repulsive couplings at half, unit and double strength in the DLMF normalization. These magnitudes are noncanonical references, not measured constants or a claim of application-specific demand for each root. Fixed-coupling radial zeros are not the energy spectrum of a fixed physical Coulomb Hamiltonian)
How they were obtained:

The attached generator certifies both a containing interval and the index of every zero. For the selected couplings $|\eta|\leq2$, write $F_\ell=C_\ell\rho^{\ell+1}\sum_{k\geq0}a_k\rho^k$, with $C_\ell>0$. The coefficient recurrence in [4] gives $a_0=1$ and $|a_k|\leq5^k$ by induction, so the series is at least $6/11$ for $0<\rho\leq1/16$. Since $(G_\ell/F_\ell)^\prime=-1/F_\ell^2$, a certified positive value of $G_\ell(\eta,1/16)$ proves that $G_\ell$ has no zero in this initial interval. For $\rho\geq1/16$, the differential-equation coefficient is at most $65$. Integration by parts and the Dirichlet Wirtinger inequality imply that two zeros are separated by at least $\pi/\sqrt{65}>1/4$. The zeros are simple by uniqueness for the differential equation. A scan with spacing $1/4$ and certified nonzero endpoint signs therefore counts every zero. The generator contains the full argument.

more

Sage/Arb complex-ball arithmetic evaluates the DLMF representation through Tricomi's confluent hypergeometric function $U$, using exact rational arguments and principal branches. The scan starts at 160 bits and increases precision when necessary. Candidate refinement uses Newton iteration; each candidate is accepted only after exact rational endpoints separated by $2\times10^{-120}$ lie inside its indexed scan interval and have strictly opposite signs. Final evaluations use 681 bits. The returned Sage interval encloses those rational endpoints. The 100-significant-digit decimals are also checked at both ends of the interval denoted by their last place. Both complex waves are evaluated, their average must have an imaginary ball containing zero, and all enclosures must be finite.

The attached checks evaluate the unsimplified DLMF phase representation independently of the generator's simplified gamma-ratio representation. They test the differential equation, the Wronskian with the regular Coulomb wave function, both derivative recurrences where applicable, and the order recurrence [5]. Derivatives are computed analytically with shifted hypergeometric parameters; the regular solution uses a separate product normalization. All 48 free-case roots are compared with Bessel zeros, and the 12 order-zero free roots with $(n-1/2)\pi$. Nonzero trigonometric controls, a deliberately reversed sign, exact decimal-interval controls and six published function or derivative values [7] test the checking methods. All 336 roots are compared with an independent calculation at 150 and 200 decimal digits. To avoid cancellation failures in direct mpmath evaluation at a zero, this calculation evaluates the Coulomb functions $1/8$ beyond the root and transfers their values with a Taylor expansion of the differential equation. Both the working precision and the number of Taylor terms increase in the second calculation. The two root estimates must agree within $10^{-140}$ and lie inside the written decimal interval. Comparisons with ball evaluations use their measured radii, and all function and derivative balls must have radius less than $10^{-130}$. The attached evidence records the actual maximum radius. The ball identities, decimal endpoint signs and interlacing with the regular Coulomb zeros are checked on the values read back from the database.



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