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Values of the elliptic alpha function $\alpha(r)$
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Numbers
$r$ 
$\alpha(r)$
1:
1/2
comment: $\alpha(1)=1/2$ [1], p. 172. It follows from Legendre's relation at the lemniscatic modulus $k_1=1/\sqrt2$, where $K=K'$ and $E=E'$.
2:
0.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727
comment: $\alpha(2)=\sqrt2-1$ [1], p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here.
3:
0.3660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878
comment: $\alpha(3)=(\sqrt3-1)/2$ [1], p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here.
4:
0.3431457505076198047932451031612076857213124984922077072932810480370700861515718445984498626894337091
comment: $\alpha(4)=2(\sqrt2-1)^2$ [1], p. 172. It follows exactly from $\alpha(1)=1/2$ by (2), with $k_4=3-2\sqrt2$.
5:
0.3318826109924715621350282485226791881971871219680396244705466526040784151965915359952943150308830019
comment: $\alpha(5)=(\sqrt5-\sqrt{2\sqrt5-2})/2$ [1], p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here.
6:
0.3260450603443938617375886288871065329291939231300498250848668090405401610425190443830740114455600463
comment: $\alpha(6)=5\sqrt6+6\sqrt3-8\sqrt2-11$ [1], p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here.
7:
0.3228756555322952952508078768196302128551295915412250901841672296005344116151418138801964432372718053
comment: $\alpha(7)=(\sqrt7-2)/2$ [1], p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here.
8:
0.3210856140096569793104537082349151734408849042842140307883951047788365633160745882704125022948886873
comment: $\alpha(8)=2(10+7\sqrt2)(1-\sqrt{\sqrt8-2})^2$ [1], p. 172. It follows exactly from the value of $\alpha(2)$ by (2), with $k_8=(1-k_2')/(1+k_2')$.
9:
0.3200403204290140825063509941091936024520117537429458560514522430711079267056925436230421476600335146
comment: $\alpha(9)=(3-3^{3/4}\sqrt2(\sqrt3-1))/2$ [1], p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here.
10:
0.3194123114290354348630863477392664394932500827422332385573279886043923450275427285250272626831183295
comment: $\alpha(10)=-103+72\sqrt2-46\sqrt5+33\sqrt{10}$ [1], p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here.
11:
0.3190256399635387920056138036820567415393655266970845517505824336540390992225554976159862801358383113
12:
0.3187824188819994232025763722448354057713686050120050926476868302362578876890595403188375317185196195
13:
0.3186265225539402003007217274235612771415265720477174234135152991436434747800514997434938771455391602
14:
0.3185249123608217165579797594738106001443891822388644606041560491233313675438422675783623943529286119
15:
0.3184576843538135943850458655255616876961516728400325511583382603514810831495870670524364842399456215
16:
0.3184125985146641503134616392721028587646304662762318000590644178259520156851045911595100344973555086
17:
0.3183819881743997790196552243721698282056770350106246658879897199510115237840919556750922157062197403
18:
0.3183609710978697902575316489833420598062783023257898876014610979656416739316373227855376758841962640
19:
0.3183463912416893984838444297707796900420870549959718175673105775856300748491025824701490439410248339
20:
0.3183361803654002477841053649110548388659954475655241670724639687346805157421530336190471400795942764
21:
0.3183289660296462534146776022547305115676992365727354989002059960441558442849703310728175777009107071
22:
0.3183238269695664434173850705722322801981286095910001685129576562943833019613393667588657332864740770
23:
0.3183201381649209847883012812447209273693428462775381647193555362752201115801033564531692823944218463
24:
0.3183174713768826509751353257546199176014499695289747453517553405633855212403230247703956616803536375
25:
0.3183155304889328333260156994666263913815478574210952646679864844804479275471399296428252393511165755
26:
0.3183141089827914374293578798563576091677272684743984800437718386154842263090147059416272746086028881
27:
0.3183130616686964459895376904241234987034534866110470018377851327120004958213216044335295928271075811
28:
0.3183122857020819454680945630865011313957762562341824678861118481480910006188512387582796139631976656
29:
0.3183117077158260571265192187450904235614088727456054749647085954909748243449065334239086610734904214
30:
0.3183112750207633726281234028693587386455879551672662324017646549422139226444298718575367886270795402
31:
0.3183109495382429344030044242393356522126191203129214503410331815525319229421003059854833968975540691
32:
0.3183107035837608841999127567747509849957480959341615671466436948420419696763264211297068496362095471
33:
0.3183105169153042655615777541016462367582198417347534382241724917497959684209610430349550032142675652
34:
0.3183103746526280109394111829469829197801042026409743075098180388045893993289940742610397700630828831
35:
0.3183102658007825709701546672295982206351445950062535159284366789508996630778736770745968481618474797
36:
0.3183101821958402023178503838388969089835671769767129693550960663888684492192712942958818778398503313
37:
0.3183101177476009905052293590853567855935847826677133718526821455748288912749706738938823132937428260
38:
0.3183100678925326027233282369121666862599781804472460348861820513592302907095669518230884217385817659
39:
0.3183100291964405176984306099696349859448739406872332675260563871357868618947750474512848544140277412
40:
0.3183099990643955771111419101445499605894339215203488208232725184138379668875443236197365526458044379
41:
0.3183099755279186306534078342708280604658649514475930219337982172229633566081732665007539007654335582
42:
0.3183099570881072322561546064700797683034838244762938661815725511931968898565133287730280800854956346
43:
0.3183099425994748873002528925168664165901482802765713454419161305693953076391853965159217512295359791
44:
0.3183099311835628584699514630042137011479858958821774558652466207800047858082275889319044435054959573
45:
0.3183099221644253572897454701762740299683230443472015437804070148804033059865064028696240259779897959
46:
0.3183099150202563701531190941563299779887537340370344868781440052626174033136603490581165272982361199
47:
0.3183099093469793536806984742121613065805070038886004846354572101924395730854872681623470237082682208
48:
0.3183099048307393669237043444750676540229977557751428226544589080851771536116814495005389142970966002
49:
0.3183099012270464299394360424281272809009398843631954274174382866347697044184885663457458836774039440
50:
0.3183098983449071683276813055952423296760864972758187919437360777934328563746453901305250591794927537
51:
0.3183098960347113617652711769994467180374712162993433022788810241711831587543205225539402401410594225
52:
0.3183098941789550307117324210568331697064319237166758571843174103688065775183065129219329415617581544
53:
0.3183098926851136703849878550271976168649642337947796547387868378262565170127310221434932814100267229
54:
0.3183098914801507487600244096557246263953682112080628610737615810124802840050501716545225244989339527
55:
0.3183098905062738770805323029052162932740958112375678779507431172772047606022876959627987472124736299
56:
0.3183098897176458889069921198416947803874480668958663746638480604675368209483849174763960292244980943
57:
0.3183098890778289614891584217018198852953698551672394063134256151002760404689501840803350232995837586
58:
0.3183098885577931049165714090782320864940323105239703787563082290217919966413316557549471857749708375
59:
0.3183098881343603868293891356586027300659179492687735486225650537664946159408066356816023887335206032
60:
0.3183098877889865027080064146048631753377112935764358069252247207156429823628397867467817812520550795
61:
0.3183098875068042148536008017243640340276702194685584022043827279353192119752862313624406055095747536
62:
0.3183098872758705967228512579360551489533373119452789314802717644614581919543589266402609421484765090
63:
0.3183098870865732927737460314331817273037992823071255832100043741706878258834458647941784452270228182
64:
0.3183098869311611510291331582271859818425327145424158673243707999819697343694562902967845265076040563
65:
0.3183098868033723644085212121845126017491675871899773301185544372054644389383243133796729772688047170
66:
0.3183098866981392365421801884206808144814158589188138605169801129527238504965346232207555426518469985
67:
0.3183098866113532966266003282477594135615449809259556952834565512884026339782408681900290653966866390
68:
0.3183098865396780493544118646491673250769459006808871927733793633323337797493392409770469213855951788
69:
0.3183098864803994050033360853778612625839899236256564241132193977588059825782690786774690023588916879
70:
0.3183098864313059774296181059384889362843028971991929932184877854025397331354641149078317165102531622
71:
0.3183098863905931056619731998162872204231466812402279336159600910351593157186337556425273165804097913
72:
0.3183098863567857562275122698074991168007631040918039807445986008671062350752511267143894808979338322
73:
0.3183098863286764811036474243785641534198033595024366176135051260038657783585321293057811553528231962
74:
0.3183098863052754038730273015987724502634709801678106160781968166766784691551929516939241340594679880
75:
0.3183098862857698332016847277046544510928776808247410934480874978348826883963392691316148406648460950
76:
0.3183098862694915959182990494999705089403649111607277289274596432186571537135341966849337662027807941
77:
0.3183098862558905709400020885425471798669083997894753012356188189761571085183747948627648096587871929
78:
0.3183098862445132126964202517587703771365723359795745601024714357792842602710662436311821259035621285
79:
0.3183098862349850961279380109495843602195621585129348713888070699307986299422669086104461400538051691
80:
0.3183098862269967084632473592292652937911778389371939957861363002974341854559424419479328492460224655
81:
0.3183098862202918664943259847288309868942941462714089166260097768521149402541538884294692922510625866
82:
0.3183098862146582603142413072142871875656471815467184220146777553682166806529554361030799582111517672
83:
0.3183098862099197220071919439069559660283203968131582614184571468139325158991810940471845430617682867
84:
0.3183098862059298957176218125146357469186351087296075560381930621289495209122000685294971269238753567
85:
0.3183098862025670479168830459875324806844970986930504805090708301616487253495469092146022475754536390
86:
0.3183098861997298067166806027144614965563025646932015097984544638977927379867922884009806681383984749
87:
0.3183098861973336592634385115056142401044269004083819719352980396661887555851913780345478777873146171
88:
0.3183098861953080685765233905974628247044367444989708350976168371324862855972240360705939189329798776
89:
0.3183098861935940972429563100211095075850132651204844042290393170430170197538963166558743719104358673
90:
0.3183098861921424464039537761884668708925335991856205526763350764202922359168350474610598234828797889
91:
0.3183098861909118354602452755198671435155240420863010975076576624456471215328084723932552547340520280
92:
0.3183098861898676616770543477949819093368134974063931004974653177136133711957141048264785600677506603
93:
0.3183098861889808900191018140213361066164646468813834283396087312805735612251531643903711694673465674
94:
0.3183098861882271325971438397517669463567942113607743439408978506565795585364804602921809059987465680
95:
0.3183098861875858844656367871068133427627699022779485733570973440426131656013328601155167333393245110
96:
0.3183098861870398885010887417372299893521187822478706434036950124503372800130579196353007549416239221
97:
0.3183098861865746069735043072772641546922540019088563769426608524264961361455721228409985672217061862
98:
0.3183098861861777814089832201288370239848002936300745093039185051774754630517640497653818248493741876
99:
0.3183098861858390655991025745315944554056789490278077770781827791061228479049796754552848548796194803
100:
0.3183098861855497192786209505871058113351337625388240523767165912835944600966277152936953896041027792
Definition
For $r\in\mathbb Z_{>0}$, let $m_r=\lambda(i\sqrt r)$. The entry is $\alpha(r)=\pi/(4K(m_r)^2)+\sqrt r-\sqrt r\,E(m_r)/K(m_r)$, where $\lambda$ is the elliptic modular function and $K$ and $E$ take the parameter $m=k^2$ [2] [4] [3].
Parameters
$r$
—   index, with $\tau=i\sqrt r$ (positive integer)
Formulas
(1)
$\alpha(r)=\bigl(\pi^{-1}-4\sqrt r\,q\,\theta_4'(q)/\theta_4(q)\bigr)/ \theta_3(q)^4$, where $q=e^{-\pi\sqrt r}$ and the derivative is with respect to $q$ [2].
(2)
$\alpha(4r)=(1+k_{4r})^2\alpha(r)-2\sqrt r\,k_{4r}$, where $k_s=\sqrt{\lambda(i\sqrt s)}$ [2].
Comments
(3)
$K$ and $E$ here take the parameter $m=k^2$. MathWorld writes them in the modulus, as $K(k_r)$ and $E(k_r)$ with $k_r=\sqrt{m_r}$.
(4)
The lattice $\mathbb Z+\mathbb Z\,i\sqrt r$ has complex multiplication by the order $\mathbb Z[\sqrt{-r}]$, of discriminant $-4r$.
(5)
The approach to $1/\pi$ is exponential: from (1), $\alpha(r)=1/\pi+8(\sqrt r-1/\pi)e^{-\pi\sqrt r}+O(\sqrt r\,e^{-2\pi\sqrt r})$ as $r\to\infty$.
Programs
(P1)
Sage
import numberdb.sage as numberdb
from sage.rings.complex_arb import ComplexBallField

r = 2
CBF = ComplexBallField(numberdb.bits(100, losing=256))
tau = CBF.gen(0) * CBF(r).sqrt()
m = tau.modular_lambda().real()
K = CBF(m).elliptic_k().real()
E = CBF(m).elliptic_e().real()
CBF.pi().real() / (4 * K * K) + CBF(r).sqrt().real() * (1 - E / K)
References
[1]
J. M. Borwein and P. B. Borwein, Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity, Wiley, New York, 1987.
Links
Similar tables
Singular values $k_r$ of the elliptic modulus —   holds $k_r$ and $m_r=k_r^2$, the singular values used to define $\alpha(r)$
Complete elliptic integral of the first kind $K(k_r)$ at the singular values —   holds the $K(m_r)$ factor used to define $\alpha(r)$
Complete elliptic integral of the second kind $E(m)$ —   holds $E(m)$ at rational parameters
Complete elliptic integral of the first kind $K(m)$ —   holds $K(m)$ at rational parameters
Data properties
Entries are of type: real number
How well the digits are known: proven
Table is complete: no (it holds $\alpha(r)$ for every integer $1\leq r\leq100$, matching the range of the singular moduli $k_r$ and $K(m_r)$; over this range the order $\mathbb Z[\sqrt{-r}]$ has class number at most $12$)
How they were obtained:

Each value is computed in Sage's arb-backed ComplexBallField at numberdb.bits(digits, losing=256) bits. The generator evaluates $m_r=\lambda(i\sqrt r)$ as a complex ball, checks that the imaginary part contains zero, and evaluates arb's elliptic_k and elliptic_e at $m_r$. The row $r=1$ is returned exactly as $1/2$. The run checks (1) on every computed row, (2) for every row where $4r\leq100$, and the closed forms of [1], p. 172, for $1\leq r\leq10$.

more

The values are proven; the closed forms in the entry comments are a separate claim, and not all of them are. $\alpha(1)=1/2$ follows from Legendre's relation. $\alpha(4)$ and $\alpha(8)$ follow from $\alpha(1)$ and $\alpha(2)$ by (2), and those two derivations were checked in exact algebraic arithmetic, with the singular moduli $k_4$ and $k_8$ agreeing with the table of singular moduli. The other seven closed forms agree with the computed balls to all 100 digits, which makes them very likely correct and does not prove them: whether Borwein and Borwein derive them or found them numerically -- J. Borwein also wrote a lattice-reduction search for such values -- is not settled here.



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