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Zeros of the regular Coulomb wave function $F_\ell(\eta,\rho)$
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Numbers
$\ell$
$\eta$
$n$ 
$f_{\ell,\eta,n}$
0
0
1:
3.141592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
equals: Bessel zero $j_{1/2,1}$
0
0
2:
6.283185307179586476925286766559005768394338798750211641949889184615632812572417997256069650684234136
equals: Bessel zero $j_{1/2,2}$
0
0
3:
9.424777960769379715387930149838508652591508198125317462924833776923449218858626995884104476026351204
equals: Bessel zero $j_{1/2,3}$
0
0
4:
12.56637061435917295385057353311801153678867759750042328389977836923126562514483599451213930136846827
equals: Bessel zero $j_{1/2,4}$
0
0
5:
15.70796326794896619231321691639751442098584699687552910487472296153908203143104499314017412671058534
equals: Bessel zero $j_{1/2,5}$
0
0
6:
18.84955592153875943077586029967701730518301639625063492584966755384689843771725399176820895205270241
equals: Bessel zero $j_{1/2,6}$
0
0
7:
21.99114857512855266923850368295652018938018579562574074682461214615471484400346299039624377739481948
equals: Bessel zero $j_{1/2,7}$
0
0
8:
25.13274122871834590770114706623602307357735519500084656779955673846253125028967198902427860273693654
equals: Bessel zero $j_{1/2,8}$
0
0
9:
28.27433388230813914616379044951552595777452459437595238877450133077034765657588098765231342807905361
equals: Bessel zero $j_{1/2,9}$
0
0
10:
31.41592653589793238462643383279502884197169399375105820974944592307816406286208998628034825342117068
equals: Bessel zero $j_{1/2,10}$
0
0
11:
34.55751918948772562308907721607453172616886339312616403072439051538598046914829898490838307876328775
equals: Bessel zero $j_{1/2,11}$
0
0
12:
37.69911184307751886155172059935403461036603279250126985169933510769379687543450798353641790410540482
equals: Bessel zero $j_{1/2,12}$
0
1/2
1:
4.457852589163553080242957544554479817865492522420599090560530576583247989424151317541804205254707971
0
1/2
2:
7.892780152719445817670364907209790025045207970289746518153890939576647060590781973275260182131755622
0
1/2
3:
11.21369054935809083586351375047028415712300848032012372426287430039770480263876532782259690074940771
0
1/2
4:
14.48544573212904646209271218612798772871871344701984714165171359760327548208029466497501431893578494
0
1/2
5:
17.72948693109060397122035161049941526285482805826263463905305103981538525366084407816922707816649717
0
1/2
6:
20.95565847284199892979384619368839106919457134815236711087156404047990783149704581805817862393400788
0
1/2
7:
24.16932281014981706172435598539687674934212286743382534970101716247394678096849699196217146219236773
0
1/2
8:
27.37373173856056027372830353556757358322492997768483567334837374583907109108061534896041045841235597
0
1/2
9:
30.57100905950356830872889994431587992253034354483826131464926750834680693873718880786199359300263056
0
1/2
10:
33.76261974819031039023417065354820431574466164093972532906169753801554804334165454999470995708890480
0
1/2
11:
36.94961766676958701509072357981312709458262017756964319781777690074515645643050458264184111958176854
0
1/2
12:
40.13278663044838948687774529993939751605816949982591399644506267341120281174088592467134396743981466
0
-1/2
1:
2.109716251219645466673968936893503213742279536770068629469528522409403288759894360014307556869871085
0
-1/2
2:
4.872418296114166492774534149420689802030855409246320269567088784800817302882756793364435182866184440
0
-1/2
3:
7.790435359523493917061473737500279747111124900714332854292136549409851430952438876036123479264058500
0
-1/2
4:
10.77484946579487425849527687539289042319041125633050293268570978487907099851424305521032294431175383
0
-1/2
5:
13.79566189539926908800093952837099918595805984770693750103265164580171966448188716146944915176151902
0
-1/2
6:
16.83934503186446863529726763392258088012420181634546659124242576787803212165444795431608985517346163
0
-1/2
7:
19.89869164757258641947193906430802453435951902242146818956611393952719133429753931795933784879422616
0
-1/2
8:
22.96942111092010127587656745538517921563147466238106835659723277238359255363876355846694175608437336
0
-1/2
9:
26.04878870432825566467459656584598599541395381825735510621759126683353492854790151470230265670018844
0
-1/2
10:
29.13493147784387708843088245968535592362655363607090886401890332419633581900210815198793997522671560
0
-1/2
11:
32.22652812293529785760506718691936393741615840043324820713242488731293243788553409215029228931115161
0
-1/2
12:
35.32260809784962554108161750794111839539555734130920382367649001325163202596591907916449681605816915
0
1
1:
5.814115615876563838170591402862290453124124345107423250426019891757680435203454347738730382347292005
0
1
2:
9.474533918374313755094179449741626758614687603472188344042103225713427851547093950884839523303414231
0
1
3:
12.94165270015528186346828067364510502657944757744505511067748926047502341955271595940575648514493699
0
1
4:
16.32324836241707365517694851803293890301884759256026129603439536092329773388549465552376488515784521
0
1
5:
19.65575667775567212927875133603984440468000682992569286932067868461431201460020665201494951505941469
0
1
6:
22.95619613675537940167782638898425486622077200346612427954017315104078202111976887191500963645854772
0
1
7:
26.23396173576931703567653150640013715531008904574618827343813058813728105335118835463805837158145936
0
1
8:
29.49481197155646957214397984459203628181332225743728423126208825465643375814980041806051787251809637
0
1
9:
32.74254115023624968914707882555073564472839176888150661439397323714553648010093717592210144205909947
0
1
10:
35.97978604117796171753512662143594149752737613256327061034647205426908739872533045182358912433013072
0
1
11:
39.20845554206361021097218747688484912432366762609894579122031074220300746830778995623385499112092335
0
1
12:
42.42997724210556838228053220501904705193371899107630104230453740179411844348201395835040020836941510
0
-1
1:
1.467395507449314329416359001261763091195603941010586748202252395711993236786056154364282199175461529
0
-1
2:
3.821871284185001239152561025905392095161516659140177994602267752150971742710510905535778140593830601
0
-1
3:
6.484727624664967673654224077536147154518108929149269244840499412662198940306545084754977338787180743
0
-1
4:
9.289374322661961124850991505195016319260485645072053559008176659876519553045109376817144467052790639
0
-1
5:
12.17310235809505696668391440669918466465010235310708949519639289314022410978780593169781767057685224
0
-1
6:
15.10665936935653041062910161448179720539668558446320213616155477986554785632545362498259471323292434
0
-1
7:
18.07428201201658110756100178405108004345980907783614535089476101207378915180977528196713083186573380
0
-1
8:
21.06657954985680789162395494764356385268821712853057350994083787478992953589639119272666023507147883
0
-1
9:
24.07753352459789715043036169574716067246523855712942791783627355553208975105228047399925360139958186
0
-1
10:
27.10306712389998323376134741670018839753843864730021531723361456081669910160386971880823636262944243
0
-1
11:
30.14029650114176345681324575539103765338656606264666524223856144312790571755484003027791017248742421
0
-1
12:
33.18710949072063262281709842376156657830739432059197606171334066009858642437540903305848209454016218
0
2
1:
8.395670123520879883126704032264995899154424953396424723645817817417813130171152805060161220897280393
0
2
2:
12.40524257859255916841542106141187210804512126862424582563478988982491543499150837121778539802276977
0
2
3:
16.11044740350688656987312337699079278683566355053769746256077794921328084555639990194240218142107077
0
2
4:
19.67631893111730249697693012385003773651559633894002851395432900285202130440010928128450019817517399
0
2
5:
23.16032851336515101409031370207507741738025967217878945000832132501742038462889310636671298717350630
0
2
6:
26.58991851300728763970562253379368259470421766548686480851811118478406101596208479987385388010846916
0
2
7:
29.98050273024073902337675723016711325272754672673982659643573476084985003566282534605137402161266464
0
2
8:
33.34166271178900006273778947966485294346086197538949807713805375879428515664310579711684332267619013
0
2
9:
36.67978647571808173915508550002274019103881742958181263969048340578591184347781479361361652929268675
0
2
10:
39.99935794640461400819859431372407673829306682950388875347894776610116181965257559561357300678994850
0
2
11:
43.30365155764794938854679685547094176818114356006763844570250664598130900155477228073124376111700035
0
2
12:
46.59513482240413986227435237452021444137328362979193377810936120384918338827143896683791993160055433
0
-2
1:
0.8560389618675377731996324554368733432470882371844282736910870889893367400594907364674483328854492189
0
-2
2:
2.546285436421290459604809149604246768002355782608348861119509432427121066062882405537315704103862557
0
-2
3:
4.704539554802972367044110785982250008028116446744046122419897666746929714523856385060621880188348978
0
-2
4:
7.128068574635721534149300409191980118349395981995848024131967211790698868595643879658500365732186469
0
-2
5:
9.713368263824748645687046994673023074787875248301545404893930395447777928501782162224628552753566554
0
-2
6:
12.40483006864877815143574797433674156496881079921831998858890520865652412006183648037478379173900995
0
-2
7:
15.17028243883276211698429531487722036694480678291378714663728635546430625497678089773854359141683201
0
-2
8:
17.98981443586225101166612663694753874862993501820302170854282188762024138323986951667793618366510332
0
-2
9:
20.85038988397862705480374301227939924443657959624009568651836773796368746689206791920692071238418338
0
-2
10:
23.74307047024174690638230179394515389521453241246203541121049593059603774992623505837252709208901030
0
-2
11:
26.66149032305194300612113382807480690253939601964228416458255592203104302109405450916885945826779627
0
-2
12:
29.60097031263569524241760278883509789000587757107682606617187370217801593601765719781174715204826729
1
0
1:
4.493409457909064175307880927280322082215583872290040802895823961926950314597104098729057809455879692
equals: Bessel zero $j_{3/2,1}$
1
0
2:
7.725251836937707164195068933062986626378159304610791186649328216729645001682688816184504845740695787
equals: Bessel zero $j_{3/2,2}$
1
0
3:
10.90412165942889982714870279018868387209485885781875858875838037854152403129767924534795971449299529
equals: Bessel zero $j_{3/2,3}$
1
0
4:
14.06619391283147347997896560060156499694237831509622273835055936239790678068662399708995660145597824
equals: Bessel zero $j_{3/2,4}$
1
0
5:
17.22075527193076873957371892506096229893332971829167936478403467046043142257400443881724600687007969
equals: Bessel zero $j_{3/2,5}$
1
0
6:
20.37130295928756284509100440543504575049474077627828310900472481472406399769752471312034311206721095
equals: Bessel zero $j_{3/2,6}$
1
0
7:
23.51945249868900654645110203911830424635003114823871066842675410735279392086189788831563784209112032
equals: Bessel zero $j_{3/2,7}$
1
0
8:
26.66605425881267352839414354427355510035094967713474434380833716343936396570339416811983469926519861
equals: Bessel zero $j_{3/2,8}$
1
0
9:
29.81159879089295883681058812719248542381985582067960421020287818369901454045657431457801550155387214
equals: Bessel zero $j_{3/2,9}$
1
0
10:
32.95638903982247672529905648511000477277068028983182458330416774436300013420899180523883684352187886
equals: Bessel zero $j_{3/2,10}$
1
0
11:
36.10062224437561069670643396053248709084302819506788073090108994274523445974627232704306794850345532
equals: Bessel zero $j_{3/2,11}$
1
0
12:
39.24443236116419284228018365930600099812761895577869058538639340151361457604383737691529311688618915
equals: Bessel zero $j_{3/2,12}$
1
1/2
1:
5.483031858162057169790087247951171188257200343001608975868928835790542209684698559473074617512104162
1
1/2
2:
8.949783179643532004996688910582255836743766579987761522565222202623077017173210469454877799735646416
1
1/2
3:
12.28434474492255548910199555521351753728407616918399498815511836549741161258770879288697934458076482
1
1/2
4:
15.56381067077782863247049511726469296975695384533173977576183054034958647607335953340632890697989797
1
1/2
5:
18.81283519924700533552879336694398005442851445074112258488723328478254825415677292375404110640622838
1
1/2
6:
22.04250178189713976398612985159975938215154419721008766236979398913288924465000169238603603581681155
1
1/2
7:
25.25875690242413878571487537249125739119216528683397959051880513858207529024238935935799051990914480
1
1/2
8:
28.46516496273900016372620630574951185332970455131545731744888271663027212940356606171199303088680188
1
1/2
9:
31.66403265014679181652693267079965288323828327730894237904995248400685532975234918490086017116467261
1
1/2
10:
34.85693924167031563937107250283242900868399885438645379285132371456396866948876946503181951691749149
1
1/2
11:
38.04501378853170776518850868172285551670184580011387911178812379186068709641975755989240026482460643
1
1/2
12:
41.22909163139540032131801836052952877643036715002112407112874583778071012648246190949554647573461856
1
-1/2
1:
3.646171915388740709314300584327127028891785094078905741651514812841469643879775358590634243406020048
1
-1/2
2:
6.616674525690986388189614762446767973888499817471680129599115383129761404985012356526688928262232710
1
-1/2
3:
9.621678390539887638550925948850509063523190746019416563335929371184656256124115760719265194461029725
1
-1/2
4:
12.65343997734354912069637333265013550827753077029462511396724492174771041436634462146487713965710254
1
-1/2
5:
15.70389972069908135248128418778643425537991390154439465644133912651108543382298061692433126572398964
1
-1/2
6:
18.76784618067264316696423813128267534488795077931143655659900141739087620639843860957133842577381480
1
-1/2
7:
21.84189890060000369822086029628830510794876359726731914121353218684132647303575787409874605424394781
1
-1/2
8:
24.92377806327502679841541650511912559070901529943670150421751911335084744021371429007358326549956170
1
-1/2
9:
28.01188475843324792972252862323550218640627929363429171552217005083387070028125131857248422491852675
1
-1/2
10:
31.10505862419956283891862463041088229319638087671482213173964811635461817746413424572824193420158674
1
-1/2
11:
34.20243274842351512877888664727318559906517945358205901445038177667443045380998344965255456761512016
1
-1/2
12:
37.30334330600056704669876545845161270530871962019779735220984564815254203667426927726440979484788358
1
1
1:
6.566570903888091781184035163508815080246723399122337350069162562872891457750070160158150289285336178
1
1
2:
10.23885719894571920529077937108994074610435845435657698256374434619461687463569153309271329069777582
1
1
3:
13.71133323647034560041977893287285778903534254161632513879927898141611899383758611948185515619146410
1
1
4:
17.09605051894979578380734260268257569642205505816672413678013924050498456781202824876471880806252795
1
1
5:
20.43062027021835133501700295682622023298435406361147550717459346888997768542198473803428420946982363
1
1
6:
23.73252890318100031572451533309570106005263770067384150659750390729055043755678035969812566341207176
1
1
7:
27.01139727641885867609992492297638836542869375030617383167735646708888574686165286325556028842753366
1
1
8:
30.27310707349288699244516412064387972194482674934020767100135839831447706491785772951090801042542846
1
1
9:
33.52152578017714498404264575051333249454673459780023829598358701326995232940287820985464200612192635
1
1
10:
36.75933648970693930755773728595253798423112790076558883383140431570441114845320058454312440603075223
1
1
11:
39.98847890433750602736994689600178885312647727637283126305067782573773309386309198463697345664691045
1
1
12:
43.21040192132869862641739015725348143297864752473731444272526496367165840161680800515607305124352804
1
-1
1:
2.966144623424196797938434664322828797989741362904768803482952750385367890338411594854414322127057869
1
-1
2:
5.661920828032430041235934505836643933830872552514354931295800357473360427363220659484534131184682798
1
-1
3:
8.478875692352065753093785028766880130896338634818320223307088544894502677155242062052573845407948310
1
-1
4:
11.36894160918253075723445113002069039416216871751403551970056482040527250164126338201319020970401505
1
-1
5:
14.30633125795444672789380724157936303833358634478414354682891846505312418342143670852959353533712557
1
-1
6:
17.27651007248916418114594267623754500134820680003067124440951320261715573445480811891053346597195832
1
-1
7:
20.27062900388119808957354445146285950094700116663874368798320972514992443784992782463807630758241432
1
-1
8:
23.28294418982210522963069753331431328904266334670426385501853334962378540409101490675637588344953161
1
-1
9:
26.30953238044039961708247097169146830022075583429894037879443056037388258616336500363680222278311613
1
-1
10:
29.34760211815584812473171984184615013176737775884276647273182506609555401116148406097641613920793656
1
-1
11:
32.39510006145816682894828604286810433867908102649333426781142237904818025679685540341504680155018829
1
-1
12:
35.45047367820516153242761903380846015520247775340792818813232129382679515719742863454049309017141604
1
2
1:
8.851065604796750199979196361369781744458192317329315393092185283602543269618520438628549504573020179
1
2
2:
12.86321976675485792183706097711613552322484380255553628378272925789303574676570591961626965824997465
1
2
3:
16.56969608246534535119429244894083670276880004234651064460282968357769161210195357104357611683815550
1
2
4:
20.13634834944015069258807209188009108175048318211097308617105993099941671826494262683192421621719260
1
2
5:
23.62089261672115067116558940413499013867436260417955460175440122977495552584109416281608390283035822
1
2
6:
27.05087419664748892803635848925004896581841153460080219942539068953204319131930602941532453940736398
1
2
7:
30.44175867618188524429414296689658264840217191072869793915324349734755920150022680933991289402268912
1
2
8:
33.80315678014002821475875238081770835428996163370459439491199074223109445304063086860923721340679817
1
2
9:
37.14147432561816998144973551188939058580735576574714637268195802983751175136440949966876097011968017
1
2
10:
40.46120675657808342209344611427510300125642904082345375565554365463495216087999893387987592313234203
1
2
11:
43.76563631319095082020496016474622373507995578808786355952368373907755873639423124835437990727632115
1
2
12:
47.05723599714351880367803622722156311985304550534813900382094756098769085628260479204064992107306216
1
-2
1:
2.048053528632148389779272480304205709096767923306865980916171953680161838646812988497411364666057893
1
-2
2:
4.223875129167604662133791246037045795172969630077008288918691637125451620280352089728810184262095505
1
-2
3:
6.653569942921671334104966538458731823222698209898525529272787831691405032459848668441561920774867403
1
-2
4:
9.241892528620435377500931846830688247203591673462677186519910149094780856396403896018522744049217672
1
-2
5:
11.93511376461681671335604719764421170453698179366923547780053085343307096121281768312727508504433846
1
-2
6:
14.70170453189218737109828054257564185114629769690790746776516170687121242518704086673431671354741359
1
-2
7:
17.52202801939485622537950439534160542715224356795961908841571932081421475902531772244941501686170504
1
-2
8:
20.38318312567078042756137919761218078851470091277292934409762274288770035945788058192031546399607264
1
-2
9:
23.27630523466551971957189939858666096534291439432026350049033228315419932756057366035262591947788849
1
-2
10:
26.19507186929293103449981190174152622871054421293418875828314885448055738102055311250086966094902516
1
-2
11:
29.13483101181944382509489060839499314169175610826688988428331348766820708049705963053894620061210640
1
-2
12:
32.09206871819495687673034472324050291221531935916657706129985089896427538876628044771225981015032882
2
0
1:
5.763459196894549791406466653952735076409087684167383429091571165635301100999911063630845210584839289
equals: Bessel zero $j_{5/2,1}$
2
0
2:
9.095011330476355156337698327989695252400929366383085415880802357627790878334597554669246146223346797
equals: Bessel zero $j_{5/2,2}$
2
0
3:
12.32294097056658205196956792532972606118942383491539433008881561514648228683067196562675293740347368
equals: Bessel zero $j_{5/2,3}$
2
0
4:
15.51460301088674823044142932724594561869071229219415021906488632871966202233513015709362042711238822
equals: Bessel zero $j_{5/2,4}$
2
0
5:
18.68903635536282220199816551630472267463713052356730917022617958741217063586752408526485882758723010
equals: Bessel zero $j_{5/2,5}$
2
0
6:
21.85387422270976579242800957037002971319992697654243228705137967378428693563738912349270018980031782
equals: Bessel zero $j_{5/2,6}$
2
0
7:
25.01280320228961246625957925460646212189196151904371787076812938614701338487067102795228798112937738
equals: Bessel zero $j_{5/2,7}$
2
0
8:
28.16782970799362387483841736024947959220971868934681966494588753422416508497033667347780051306420615
equals: Bessel zero $j_{5/2,8}$
2
0
9:
31.32014170744717453576203217405926081851077509080265596807813532428522217451612040745049117588895704
equals: Bessel zero $j_{5/2,9}$
2
0
10:
34.47048833128498866573412721373850971870762093242062396197530466065179024271706372349342728034190162
equals: Bessel zero $j_{5/2,10}$
2
0
11:
37.61936575358842484806240968535259406369585035388744261422669888240807720173114724740036373304550080
equals: Bessel zero $j_{5/2,11}$
2
0
12:
40.76711582140680476154142762746564360940599772175243059384183354014839800423716942169381542460566150
equals: Bessel zero $j_{5/2,12}$
2
1/2
1:
6.624638311280935725383169745695093018548321507733237864506841952731458159764681789315338714338016476
2
1/2
2:
10.15138905519545681232313743448718705962444236690230453549225520778195960224680295559181796952008826
2
1/2
3:
13.51545754254780589267084104146589482095397861476335119498746859960677648731004259023326469634937281
2
1/2
4:
16.81282679219249776226158525017355679059190402063122207787767409131572225643163002032960908026007982
2
1/2
5:
20.07395480169401562088484084648584860154653701711026954731856863880742096602964082635965197582841681
2
1/2
6:
23.31237895341969440380587630896316912379214201455959003014651831824306531036548978276084868514388604
2
1/2
7:
26.53527656462093609585794610881583013857330653657275323202529358047848055141605709377867319021770567
2
1/2
8:
29.74690140133999492667965096907148180763421128105264366205400634112781466854330262957162094753210983
2
1/2
9:
32.94997768189349684477555168895366325006408768898766481499961472144358204533364008339406003488474974
2
1/2
10:
36.14635289839935674306660806843712020729394440203771670103593619527084246295412205546626359475379128
2
1/2
11:
39.33733651748519592803116940056553930496824461177360183115321861151915884103542677032018377773033260
2
1/2
12:
42.52388982426367256924848161772436983589205400349637709196348223843228655286062395890565159657429003
2
-1/2
1:
4.995696239514856050253527373922222971416122014495959648672672617735993074947148977670040490931763115
2
-1/2
2:
8.120198516639528414709183494985347501876928545125966221152387013951934725247767726623645795428072025
2
-1/2
3:
11.20238010736849073076066653036292623840951170167431292772336863059427215362233111850451488225953698
2
-1/2
4:
14.28080092987333238739741601018702898571814605658509056259949040487675060965742125797186897269742945
2
-1/2
5:
17.36253483168815641278210014501170600044573286166087030399030154553083982393297262488867183973757657
2
-1/2
6:
20.44889794104748793976285659029585099707334761829035120979885763789521011870945546952354346834272472
2
-1/2
7:
23.53980040503010221246032727424768410557859139405300755402625546489326031702014032813809926319230361
2
-1/2
8:
26.63480310940214835795822337924470257498078795548970187018571206370083187542426475814472722863119862
2
-1/2
9:
29.73341748999181885425397485544690117193177740330412659721982190870279501021255497155294190465987960
2
-1/2
10:
32.83519267265411783110603254866765877190018067959602112100763213434451958574314768786606582692343368
2
-1/2
11:
35.93973622405529604162726943804296163907665785379759316875470895849840242151077591716339153786760214
2
-1/2
12:
39.04671359637581992872719189571003998618114245170463442554978341336080946181627607447351527449853550
2
1
1:
7.562500669172140241201706926037044791487289072451136074115374514393377517411034215140976392861279136
2
1
2:
11.26984344211908912682053615510124789348534325250231270463524854557153859222990851650870586668635899
2
1
3:
14.75976803358200508545444276492307846815383210252642857074714239777812550308519793477012582027640379
2
1
4:
18.15519257073704280354527802401370840120775597610769075046814348216052943179470897342157790406034085
2
1
5:
21.49706977861219293072884245884510935896095631904450047660517467251136145561944608912371368844912144
2
1
6:
24.80430761374820570966508486688793835144154765941620906810811236329839513608666756532418610615838708
2
1
7:
28.08724499162449080548508663885873569605334430042956394519448803389369753390100902542420452586979294
2
1
8:
31.35216853485848641603170761021478090149743513109450330461760652170981029355543654569165013344953886
2
1
9:
34.60319255975207020761391935657524287007507724702417401861832436881533654833637400192978272119037997
2
1
10:
37.84315964607807246226681638813226178129265528254121548978946539825147945377217850400747970596023589
2
1
11:
41.07411734502876618526454792092523421235081982796476062198981418012457985776832092620023002195144033
2
1
12:
44.29759020472294741964868996428711690986638322207676089881949860583622970388308304100208481055451557
2
-1
1:
4.330664566970901897851353408542084963408434003774106104224600039731809897855393399177413887522492147
2
-1
2:
7.241596516744938158583461560832106163626405988307103410046902950138534616289156720169690790364228824
2
-1
3:
10.17046439366167848143526025786534574422012523398363257316368859636958659316777159621509590358595052
2
-1
4:
13.12903020023453034819109626089026490062972563662826350196047499401575487825806969649118461135179969
2
-1
5:
16.11251392060502026412650939654087657471253173150585187726171768562046430703421757244207655294518349
2
-1
6:
19.11574982555410552555832133563685654587820814346838301513400511061508440517740733913181401459883011
2
-1
7:
22.13469327461966844130361088523937193467094992169822125468157953869039239931724979138358316638299056
2
-1
8:
25.16631259176776840998361321002464106981823618313308724867889502687671887178687780316941294515690752
2
-1
9:
28.20832856964777217375419414195331316483816270222516034282651766236121254642735927351642580666444003
2
-1
10:
31.25900283357344826482304888336283123465896543585575629275543758525427150346625434995474191185202081
2
-1
11:
34.31698700993261403396991031810238380656179789662501021679589008823122957191996221357687020617939898
2
-1
12:
37.38121785467869108302424076091371315944506781566472308980061042975960294160336086085755745143454372
2
2
1:
9.593002170650343462868855132183945822290388672234487222581848084798790158049963592411162853536044984
2
2
2:
13.61773053514099982462994758051206004357247754204800756118161974826224900232157016164226127236061328
2
2
3:
17.33071659280815198996779951279913018421740579358924920014370235340200878549552872138696170858147623
2
2
4:
20.90147786942332656885808624497639865494632191668054518234383076407034490026738678617429154608509727
2
2
5:
24.38888743974486200180174048765789093285224333437000120773305784871876475840055994935861755397341208
2
2
6:
27.82099467305180571181745522814508006319948868123279805372574915524001399290751485408427940603667686
2
2
7:
31.21352502786555694081926701190203446144970016376692586540025874259247744492154735947542115050431083
2
2
8:
34.57623838467987000959925859410342459101446475567126760370485363739952952990745207483308088343655579
2
2
9:
37.91563287960693470587085856857588339041824320882828467882021793272272558334151896237435743972841721
2
2
10:
41.23626441205301954787301154884391092440872921861888973698307752219126771639293886644765975316576260
2
2
11:
44.54145659067895921265513302549574079405272770990813449633060925148710787595313499568872468025482608
2
2
12:
47.83371174286397157080255513268614561843942980634413575013003002300376379894845621446123039718343250
2
-2
1:
3.299211545608703054046774754894839007665676755174033277340809417835614601133165458351818957143444641
2
-2
2:
5.790855620565750277479653366819642655953170072735539858559865939481133563789851110796781158132601170
2
-2
3:
8.403866695254212915130199785128135968982670068706553118811957277741375366915791749999269779608230559
2
-2
4:
11.11018362995749645211894320448966982627970612798755989981827573705837405843383211629793759911791739
2
-2
5:
13.88481419277625626132768467111944054849260939536294384651295477377066624222742962585721686939620017
2
-2
6:
16.71054273875829583732097831642947706025447559620071432254121667092997324473141971444409069239928033
2
-2
7:
19.57556525885665333550667395628876861051068804936276346362948635884037587757247085383756919613374176
2
-2
8:
22.47158344707950561693211549467306505060822691428782275388776619276255590334979876834638418517697526
2
-2
9:
25.39259543120591953412698521106060574043435602719484550101687186498150345835963751179191378217553426
2
-2
10:
28.33414371874150984053914008354927510651013591922618642407933327529970064134938613926680299149228161
2
-2
11:
31.29283895069150509184931524075829908339963727742564076484870821697727767605864990104577290007550944
2
-2
12:
34.26605021094890928811438820584583167958070187035756853513663219501325438794712342727249030731745313
3
0
1:
6.987932000500519959015333385644185528506770621503095183519825748258402009511698649709701547408971964
equals: Bessel zero $j_{7/2,1}$
3
0
2:
10.41711854737936476325043101974306030925969211863415879169469582062819956886924741158793893226811179
equals: Bessel zero $j_{7/2,2}$
3
0
3:
13.69802315324924899950308661888158270875702064242712375512483292498710988503083891601146895390353060
equals: Bessel zero $j_{7/2,3}$
3
0
4:
16.92362128521383957885418342369601906036197683809949406627571215958164381850695145162521271019769681
equals: Bessel zero $j_{7/2,4}$
3
0
5:
20.12180617445381828564964637682951749971199782660683905514600901300105123923208690623146932878259274
equals: Bessel zero $j_{7/2,5}$
3
0
6:
23.30424698893965135202747637355163305529941691681748209521124134219361301254616217124447661738355354
equals: Bessel zero $j_{7/2,6}$
3
0
7:
26.47676366453912815003998633931609343521963310429559739531137063788717025444105266397212356442698178
equals: Bessel zero $j_{7/2,7}$
3
0
8:
29.64260454031580917207262759862798631148815192682010530372972496262958039607580416959491048502937697
equals: Bessel zero $j_{7/2,8}$
3
0
9:
32.80373238519610794296401521024579883157440531018071287264972459766613480059940676565243359119776409
equals: Bessel zero $j_{7/2,9}$
3
0
10:
35.96140580470903306894817567629972776499244979006996561538505974476053074334518883553107300175190029
equals: Bessel zero $j_{7/2,10}$
3
0
11:
39.11647019027153538632076425884314006076758533121464372523902151987284090502116001201027778085968248
equals: Bessel zero $j_{7/2,11}$
3
0
12:
42.26951497778116151223771160714448484212186589053469751387149926125683718219846308425322515873865696
equals: Bessel zero $j_{7/2,12}$
3
1/2
1:
7.779464239711966084109937863388276305634727410537810759557668567162242463595098612369780467481379882
3
1/2
2:
11.37664257978539775498167103464895983048691515642208880200779366943974912376498076679926540333029021
3
1/2
3:
14.77863972186042614671605381252953007230223313058269319533317545563523940907996576870064992789296772
3
1/2
4:
18.10029791504493536379575958188086369294891001635879723764452463514657065546093007379734402437025669
3
1/2
5:
21.37845574885767784412587579637598375784354822192813132901114958801405055565035466166642458962304820
3
1/2
6:
24.62953216661862069260928595143680102501871512839534447028318056410639536209969147074827558809043921
3
1/2
7:
27.86222152102430903235791213900483982806329026460403144018538807662872160399408594141597650869481002
3
1/2
8:
31.08165947674533801992873300849763222763230195876781325544211404284501049819867036097134341478592776
3
1/2
9:
34.29112043863752699257611082941219322462468664201344857503162729168958035525556807724955993160909782
3
1/2
10:
37.49281395393892034521026902014914063607538735852369070102110058762782643368851320494474561019252471
3
1/2
11:
40.68829795533945989191588242673127325885631133024490758668784964523635624626315641717528431000631564
3
1/2
12:
43.87871013805350225429793186692482063245322865100346732783377419521848786886246079497379284739060860
3
-1/2
1:
6.265527576202704248849723872845161341183580110572870614972277066487018991003421777631452613183330697
3
-1/2
2:
9.520329219083893900701016703397144836734611657590764136819944390837297046271440845376730746304483278
3
-1/2
3:
12.67434175787275687623515556489482679202722702915069627136069560649481549026919627800288307606362065
3
-1/2
4:
15.79900366358925399895225810515704931034787032358710605873289523627970819228859953547425765432216723
3
-1/2
5:
18.91313374042412774070855118848541139450239607578765962344046254744877334577784257610645427950030388
3
-1/2
6:
22.02349191475133610291708300348903612768566337009338162233541535295295721751541489959898542459168633
3
-1/2
7:
25.13289182193067971417400635927281024845948315490712761298050026515109529559861928897798896951580253
3
-1/2
8:
28.24259288615267529774604267165795548199259415865113440172867732778149042465876872301591646552124408
3
-1/2
9:
31.35316891850632993297018291601889331761881655582612091496510817709490550507028847658551608583926434
3
-1/2
10:
34.46486842478280994272456928872756468724497050372194153352286834237297952020054025843078729908982846
3
-1/2
11:
37.57777850212596784280585503194105884545958065159260238449705457874327224198313416086775891109196357
3
-1/2
12:
40.69190449964209010549123336244491735801321556543762077951053665701061280815759926090292635955738166
3
1
1:
8.632432707520534310175902426017675579107730014900420669082660991149493090695382622247832743848290969
3
1
2:
12.38935043900727095943711313730610754234507076931881584129428823898256184856824264160915966792095290
3
1
3:
15.90596717981943620228389039024443028712908691091809408647416941416034988436835004165306303120503594
3
1
4:
19.31854153400749088865619483680621711850669350417552055046806954089215731184300651967767358860651634
3
1
5:
22.67249042282982792789188785429490191561467529333207942456450848686747273822671012026911725082804156
3
1
6:
25.98873143881572655665710963348417118172591044844840233191369013974002444381730843540380574900332322
3
1
7:
29.27866138542988995061683965921961107585918345814271922595160653691676909501146505028334435685973347
3
1
8:
32.54918266572385407340022505816212574677073617560184334123703055714263046369192125024017448793097242
3
1
9:
35.80479450956637491036411681402536652559424527632270024355216422037145831038519945361467366473796800
3
1
10:
39.04859335892608018525502158207461436651405101975420942178444182364089585887670721300555778887291701
3
1
11:
42.28280138855960449181685435328345511965583931437402936601842265063793744587006652971354834479677547
3
1
12:
45.50906741786342761709539439689946411469603690939551029858999484188200765003573176027838731663673678
3
-1
1:
5.616769950220125610052253671789932765716974098586605201522734787993791893288499884279717856524562710
3
-1
2:
8.693565431077474436360503604670308446103210674763466242904560160482951990939711303979524295632151980
3
-1
3:
11.71611799293020590156150143905026039177191158724059884165683305284589060024663286391099228060642318
3
-1
4:
14.73559176020575999759208640106922759452742844438866603666758618239246720445055356305704575257145600
3
-1
5:
17.76192628128566054459696335902910203332756309676167059791562851705635537678105585882088612657234944
3
-1
6:
20.79695011028851394990711659805086574381797446180386225858395270094139583819867050252053524334263525
3
-1
7:
23.84040405747420930713879940970247487290302464020876520388428798198685533934089198770517906526580869
3
-1
8:
26.89149080020438152341025507508076079628656099107378720078927216836459517513220935037179979798800931
3
-1
9:
29.94933641457725814255481117300067872974809231868450965846380497758175781620402603492426890629359049
3
-1
10:
33.01313084019550379656729519034527469186571151873839389750702074555829978761288169082434363624458710
3
-1
11:
36.08216353364550992906356541009665061691459684109558635983475602598133968821415585029228407405228657
3
-1
12:
39.15582389082247979464133749798396100192361854015954392553615473682683591262741328429293804453622100
3
2
1:
10.48424512460042602286770019148782602998358940606419942026887430097010891860280883162527003541412538
3
2
2:
14.53309038596434399425866926369653222687526982759631023784414521851239055521194309582622893173206744
3
2
3:
18.25924303293078110949706627514747672820024110065592493450112174436960895415118133637039423943514593
3
2
4:
21.83857779738468091391602031346380222430157314896036078594365857644971513063923758194850245407993334
3
2
5:
25.33209309414184926161975630742688783513019594398579724541347358146445655886025282324536541024748316
3
2
6:
28.76880015113853150362139709670336154291253389438326144881329396014659128954256165703237059216405266
3
2
7:
32.16493440300115629544397941461389382585744840811068082886941175570721934538418076589173819847853727
3
2
8:
35.53055484224036253943803195140058173244164238958424942796557010090397307301003912953608083386234737
3
2
9:
38.87234797425965438800211587471552868782744552765827974248632373934086542642877908770131665202272409
3
2
10:
42.19499481952354563889560158264261033143162620472996236224608095601140834962805265642065869255893288
3
2
11:
45.50190562987156581173277438301419317501931168510751431242652654768025062003043159968488553714102387
3
2
12:
48.79564477807136624163789120772279872250673569483589963150125511254820451194375275087207742631505232
3
-2
1:
4.539280913041982737207014685093625673671981292080857317033888017348367548747198120761336705289017493
3
-2
2:
7.262734829920972291468345260757338924580386046224898492666043848266467733221489540068477623590115233
3
-2
3:
10.01580012006367529522360211208757612935573647378455196018145738263732539099540214076642057742286659
3
-2
4:
12.81625788072491230680200640890152348835324833728268173275899292045426291111551488722969386904305109
3
-2
5:
15.65831517473727046840623493970906334626411580049527137767133369173659876720677329306005936123040415
3
-2
6:
18.53456111700980195174157895158099625884545668795745812062493372077459323575131246635497617839685990
3
-2
7:
21.43874953605167423403497073887231231410880369152430488681855859413035967178522713191266585346099433
3
-2
8:
24.36596461557810175348956958041677682974990035758511525862024027303699836535640313525002718838428883
3
-2
9:
27.31237659632904401611930425329845006409499917468160187932431246767003677702537971755905819450354811
3
-2
10:
30.27498370148191724025026774812126910566822520232687492366627101506581318740081663509829904338069271
3
-2
11:
33.25140696336865271258798199357816070364112948774010035073009193792142652739129710636529383037966946
3
-2
12:
36.23973766519274021012387147173892296274816413741059676196722556974927128259891654186892364457477711
Definition
The number $f_{\ell,\eta,n}$ is the $n$th strictly positive zero, in increasing order of the dimensionless radius $\rho$, of the regular Coulomb wave function $F_\ell(\eta,\rho)$ 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)
$\frac{d^2F_\ell}{d\rho^2}+\left(1-\frac{2\eta}{\rho}-\frac{\ell(\ell+1)}{\rho^2}\right)F_\ell=0$ [1].
(2)
$C_\ell(\eta)=\frac{2^\ell e^{-\pi\eta/2}|\Gamma(\ell+1+i\eta)|}{(2\ell+1)!}>0$ [1].
(3)
$f_{\ell,0,n}=j_{\ell+1/2,n}$, where $j_{\nu,n}$ is the $n$th positive zero of the Bessel function $J_\nu$. In particular, $f_{0,0,n}=n\pi$ [2].
Comments
(4)
The regular Coulomb wave function satisfies $F_\ell(\eta,\rho)\sim C_\ell(\eta)\rho^{\ell+1}$ as $\rho\to0^+$, with the positive constant in (2). Multiplying this solution by a nonzero constant does not change its zeros.
(5)
Positive $\eta$ describes repulsion and negative $\eta$ attraction. In scattering, $\rho=kr$, where $k$ is the wave number and $r$ the separation [5]. These zeros are radial nodes at fixed $\eta$, rather than energy levels at fixed physical charge, because changing the energy also changes $\eta$.
(6)
At $\eta=0$, the zeros coincide with the positive zeros of Bessel functions of the first kind of order $\ell+1/2$. In particular, $f_{0,0,n}=n\pi$ [2].
(7)
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. [7] [8] [9]
Programs
(P1)
Python
import mpmath as mp

mp.mp.dps = 150
# First positive root for angular momentum 0 and Coulomb parameter 1.
root = mp.findroot(lambda rho: mp.coulombf(0, 1, rho), (5, 6))
mp.nstr(root, 100)
# The attached generator additionally certifies the interval and root index.
Links
[7]
DLMF 33.2
[8]
DLMF 33.5
Similar tables
Values of the regular Coulomb wave function —   the values of $F_\ell$ whose positive radial zeros are listed here
Values of the irregular Coulomb wave function —   the companion solution $G_\ell$ of the same radial differential equation
Zeros of Bessel functions of the first kind —   orders $\ell+1/2$ give the free case $\eta=0$
Data properties
Entries are of type: real number
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 well the digits are known: proven
How they were obtained:

The attached generator certifies both a containing interval and the index of each zero. For the selected couplings $|\eta|\leq2$, write $F_\ell=C_\ell\rho^{\ell+1}\sum_{k\geq0}a_k\rho^k$. The coefficient recurrence in [3] gives $a_0=1$ and $|a_k|\leq3^k$ by induction. Consequently the series is at least $2/5$ for $0<\rho\leq1/8$, so this interval contains no positive zero. For $\rho\geq1/8$, the coefficient $q(\rho)=1-2\eta/\rho-\ell(\ell+1)/\rho^2$ is at most $33$. Integration by parts and the Dirichlet Wirtinger inequality imply that two zeros are separated by at least $\pi/\sqrt{33}>1/2$. 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 full argument is included in the generator.

more

Sage/Arb complex-ball arithmetic evaluates the DLMF hypergeometric representation at exact rational arguments. The scan uses 160 bits; final endpoint checks use 681 bits. A 150-decimal-digit mpmath solve supplies each candidate, which is accepted only when exact rational endpoints separated by $2\times10^{-120}$ lie inside its indexed scan interval and have strictly opposite signs. The returned Sage interval encloses these endpoints. The 100-significant-digit decimals are independently checked at both ends of the interval denoted by their last place. Each complex enclosure of the real regular solution must be finite, with its imaginary part containing zero.

The attached checks compare all 336 roots with mpmath at both 150 and 200 decimal digits [6], and all 48 free-case roots with Bessel zeros. They test the differential equation, the Wronskian with the standard irregular Coulomb solution, and the regular-function derivative and order recurrences [4] at every applicable entry. The full regular function uses a separate product normalization and analytic derivatives; the irregular function uses the DLMF representation through a separate confluent hypergeometric representation. The largest measured radius among these function and derivative balls is less than $1.14\times10^{-154}$. Controls include nonzero sine and cosine values, the unit Wronskian and a deliberately reversed sign, exact decimal-interval parsing, both normalization formulas, and eight published function or derivative values in the mpmath documentation. The checks are repeated on the numbers read back from the database.



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