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{
"T295": {
"Title": "Singular values $k_r$ of the elliptic modulus",
"Numbers": {
"1": {
"modulus": "0.7071067811865475244008443621048490392848359376884740365883398689953662392310535194251937671638207864",
"parameter": "1/2"
},
"2": {
"modulus": "0.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727",
"parameter": "0.1715728752538099023966225515806038428606562492461038536466405240185350430757859222992249313447168545"
},
"3": {
"modulus": "0.2588190451025207623488988376240483283490689013199305138140032073150569747488019969223679746942496655",
"parameter": "0.06698729810778067661813841462353190826429868654740484298604825513701674577279999072971345331068785608"
},
"4": {
"modulus": "0.1715728752538099023966225515806038428606562492461038536466405240185350430757859222992249313447168545",
"parameter": "0.02943725152285941437973530948362305716393749547662312187984314411121025845471553379534958806830112718"
},
"5": {
"modulus": "0.1188769458026001011927468428418141369035869990099698731574444523370427533163632995562193805442976604",
"parameter": "0.01413172824335432181713612410546743780751401599762698904826158756188447208624290188152867417004787703"
},
"6": {
"modulus": "0.08516423317474258764879930929823996871221423752353523819094705017336145435070102300137673796385424070",
"parameter": "0.007252946612241925929625505032827059292755203645759397066478181225473860512157819228345025253462944667"
},
"7": {
"modulus": "0.06262291254316797010266468003906811674412447729401465915328671833152028488492276107179408847356814334",
"parameter": "0.003921629175389264280947046192638670179326403172040591180937288899799595644321819794926333786023073010"
},
"8": {
"modulus": "0.04702189950099110167091430362349100744278102015792997239838804325672867247235649994365432741515549097",
"parameter": "0.002211059032681307216329011198714649106853496189276288763252159574974218430159353419724044688621803427"
},
"9": {
"modulus": "0.03592156820389893411062550016477818118301192441180663807703476337756399733968415297849160326424456135",
"parameter": "0.001290359062227362895139482184855380406431958841196462290857508300142681403283012349239937518313465227"
},
"10": {
"modulus": "0.02784244474454950317011830363123913081422323179123602254900400468009448413913729947741189758862122209",
"parameter": "0.0007752017293532922488376453553255501692319869434272712607280923833754612812020806740099251537389029166"
},
"11": {
"modulus": "0.02184972261085453699635444400253632837557616482636254693682044065139242811863149389527736992693722323",
"parameter": "0.0004774103781712880047613845988705322979509030521801237358010434701145669317780402524325961119304859579"
},
"12": {
"modulus": "0.01733238012099927101897904171464979805003839277354352208350305220237708590543974890613755215373146914",
"parameter": "0.0003004114006588107046886875131926651573156840693907717841913330506224450179699313700922032483917177902"
}
}
},
"T296": {
"Title": "Complete elliptic integral of the first kind $K(m_r)$ at the singular values",
"Numbers": {
"1": {
"number": "1.854074677301371918433850347195260046217598823521766905585928045056021776838119978357271861650371897",
"comment": "$K(m_1)=\\Gamma(1/4)^2/(4\\sqrt\\pi)$ CITE{MathWorldSingularValue}.",
"equals": "HREF{Complete_elliptic_integral_of_the_first_kind_K#1/2}"
},
"2": {
"number": "1.645568395293458039866051685287072715999557026055401037265292137149578863729330871593184129832048067",
"comment": "$K(m_2)=\\sqrt{1+\\sqrt2}\\,\\Gamma(1/8)\\Gamma(3/8)/(2^{13/4}\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"3": {
"number": "1.598142002112540144460965105390112888900611097136742843395001850717051182376299048153673076745023371",
"comment": "$K(m_3)=3^{1/4}\\Gamma(1/3)^3/(2^{7/3}\\pi)$ CITE{MathWorldSingularValue}."
},
"4": {
"number": "1.582551727223715911833135071070409876529488149618789243497169448478208535186663551736209814065543222",
"comment": "$K(m_4)=(1+\\sqrt2)\\Gamma(1/4)^2/(2^{7/2}\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"5": {
"number": "1.576390394802592476323757555260863247741937034744074005958674897925807012418125362905820159225379507",
"comment": "$K(m_5)=(2+\\sqrt5)^{1/4}\\sqrt{\\Gamma(1/20)\\Gamma(3/20)\\Gamma(7/20)\\Gamma(9/20)/(160\\pi)}$ CITE{MathWorldSingularValue}."
},
"6": {
"number": "1.573656231264378266467658770764262615447771495128054395083197322369080422472509636815994782912924097",
"comment": "$K(m_6)=\\sqrt{(\\sqrt2-1)(\\sqrt3+\\sqrt2)(2+\\sqrt3)}\\sqrt{\\Gamma(1/24)\\Gamma(5/24)\\Gamma(7/24)\\Gamma(11/24)/(384\\pi)}$ CITE{MathWorldSingularValue}."
},
"7": {
"number": "1.572339753406137527416490497176982668311869155892281594080158998982190209092426217240368536444070701",
"comment": "$K(m_7)=\\Gamma(1/7)\\Gamma(2/7)\\Gamma(4/7)/(4\\cdot7^{1/4}\\pi)$ CITE{MathWorldSingularValue}."
},
"8": {
"number": "1.571665689206437042031861058437591105397806921254527952296192373178656438735452990723968471906921488",
"comment": "$K(m_8)=\\sqrt{(2\\sqrt2+\\sqrt{1+5\\sqrt2})/(4\\sqrt2)}(1+\\sqrt2)^{1/4}\\Gamma(1/8)\\Gamma(3/8)/(8\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"9": {
"number": "1.571303417736683352208481497338796809391174821642956915852621765840171701094284147643588839960207742",
"comment": "$K(m_9)=3^{1/4}\\sqrt{2+\\sqrt3}\\,\\Gamma(1/4)^2/(12\\sqrt\\pi)$ CITE{MathWorldSingularValue}."
},
"10": {
"number": "1.571100880616723881003002373510769622686969652665615951839918153838543584392595612006151433598168530",
"comment": "$K(m_{10})=\\sqrt{2+3\\sqrt2+\\sqrt5}\\,\\sqrt{\\Gamma(1/40)\\Gamma(7/40)\\Gamma(9/40)\\Gamma(11/40)\\Gamma(13/40)\\Gamma(19/40)\\Gamma(23/40)\\Gamma(37/40)/(2560\\pi^3)}$ CITE{MathWorldSingularValue}."
},
"11": "1.570983855774829120451912490395857958850098334483956144147357301154551652926723296219209863525485519",
"12": "1.570914318015180699495529303260880207317602201733998524304216557139476320543522033579062257357751935"
}
},
"T297": {
"Title": "Ramanujan's class invariants $G_n$, with Weber's $\\mathfrak f(\\sqrt{-n})$",
"Numbers": {
"1": {
"ramanujan": "1",
"weber": "1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674"
},
"2": {
"ramanujan": "1.023807274909184034748664952421301398266836573959224042897868368355500754030548719031269015427436305",
"weber": "1.217518895713548480868409321542186128270770427496416662828070102569571075237846008749157491208572913"
},
"3": {
"ramanujan": "1.059463094359295264561825294946341700779204317494185628559208431458761646063255722383768376863945569",
"weber": "1.259921049894873164767210607278228350570251464701507980081975112155299676513959483729396562436255094"
},
"4": {
"ramanujan": "1.094588288578459130320957082252498229478023913160772989188705492843845419652414570169519693428444374",
"weber": "1.301692180776155281275810494127103187985637798553643888563039263015085924275580170090954315998461252"
},
"5": {
"ramanujan": "1.127838485561682260264835483177042458436468335423655767290049186400476209539553612403996109293393067",
"weber": "1.341233551603846239042001605587290253445981548244120118800287006706080096468417964112692309432044359"
},
"6": {
"ramanujan": "1.159287416192176064416754909845814112231606704790906857611078700657237474707582791869945271633129378",
"weber": "1.378632843668856481541320330743740452980424002187762797029736783442185100551146541257711564863959260"
},
"7": {
"ramanujan": "1.189207115002721066717499970560475915292972092463817413019002224719466668226917159870781344538137674",
"weber": "1.414213562373095048801688724209698078569671875376948073176679737990732478462107038850387534327641573"
},
"8": {
"ramanujan": "1.217855815652096293978713299458821102467272732345996841867457044017639413767180434851854965124016807",
"weber": "1.448282801020915144391302213193110660388445736120313511631803289407580167884474347933537540743048732"
},
"9": {
"ramanujan": "1.245445108413552301377364874341648400757800564633019580694266561964799501770347871190061384491314176",
"weber": "1.481092184270731698455563868246081430655643995054625857304393098740749362747188178747915199145171498"
},
"10": {
"ramanujan": "1.272142962269806534370186786345239532637330320472426116129544419548486973789089432973820206159163730",
"weber": "1.512841462031892066221052361565299056747656895155674137146748613298590500767155829815710386698019804"
},
"11": {
"ramanujan": "1.298082557039312863788515964642505646858690843920203858872931594728899234940826443528624428193841196",
"weber": "1.543689012692076361570855971801747986525203297650983935240804037831168673927973866485157914576059125"
},
"12": {
"ramanujan": "1.323370411329764949923802090511931770532358163647765022928671844033190145004186076998826343392241821",
"weber": "1.573761508937434068908197823251603868609915083857076387667756697723708085998064730825060787314332664"
}
}
},
"T299": {
"Title": "Values of the elliptic alpha function $\\alpha(r)$",
"Numbers": {
"1": {
"number": "1/2",
"comment": "$\\alpha(1)=1/2$ CITE{BorweinAGM}, p. 172. It follows from Legendre's relation at the lemniscatic modulus $k_1=1/\\sqrt2$, where $K=K'$ and $E=E'$."
},
"2": {
"number": "0.4142135623730950488016887242096980785696718753769480731766797379907324784621070388503875343276415727",
"comment": "$\\alpha(2)=\\sqrt2-1$ CITE{BorweinAGM}, p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here."
},
"3": {
"number": "0.3660254037844386467637231707529361834714026269051903140279034897259665084544000185405730933786242878",
"comment": "$\\alpha(3)=(\\sqrt3-1)/2$ CITE{BorweinAGM}, p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here."
},
"4": {
"number": "0.3431457505076198047932451031612076857213124984922077072932810480370700861515718445984498626894337091",
"comment": "$\\alpha(4)=2(\\sqrt2-1)^2$ CITE{BorweinAGM}, p. 172. It follows exactly from $\\alpha(1)=1/2$ by CITE{formula-order-two}, with $k_4=3-2\\sqrt2$."
},
"5": {
"number": "0.3318826109924715621350282485226791881971871219680396244705466526040784151965915359952943150308830019",
"comment": "$\\alpha(5)=(\\sqrt5-\\sqrt{2\\sqrt5-2})/2$ CITE{BorweinAGM}, p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here."
},
"6": {
"number": "0.3260450603443938617375886288871065329291939231300498250848668090405401610425190443830740114455600463",
"comment": "$\\alpha(6)=5\\sqrt6+6\\sqrt3-8\\sqrt2-11$ CITE{BorweinAGM}, p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here."
},
"7": {
"number": "0.3228756555322952952508078768196302128551295915412250901841672296005344116151418138801964432372718053",
"comment": "$\\alpha(7)=(\\sqrt7-2)/2$ CITE{BorweinAGM}, p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here."
},
"8": {
"number": "0.3210856140096569793104537082349151734408849042842140307883951047788365633160745882704125022948886873",
"comment": "$\\alpha(8)=2(10+7\\sqrt2)(1-\\sqrt{\\sqrt8-2})^2$ CITE{BorweinAGM}, p. 172. It follows exactly from the value of $\\alpha(2)$ by CITE{formula-order-two}, with $k_8=(1-k_2')/(1+k_2')$."
},
"9": {
"number": "0.3200403204290140825063509941091936024520117537429458560514522430711079267056925436230421476600335146",
"comment": "$\\alpha(9)=(3-3^{3/4}\\sqrt2(\\sqrt3-1))/2$ CITE{BorweinAGM}, p. 172. The closed form agrees with the value here to all 100 digits; it is not proven here."
},
"10": {
"number": "0.3194123114290354348630863477392664394932500827422332385573279886043923450275427285250272626831183295",
"comment": "$\\alpha(10)=-103+72\\sqrt2-46\\sqrt5+33\\sqrt{10}$ CITE{BorweinAGM}, 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"
}
}
}