Table of constants with 50 decimal places

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Here are given the first 50 decimal places of constants that occur frequently in numerical computations (see also [1], [2] and [4] for large tables of constants available on the net). This number of digits should be enough for most practical purposes.

1   Related to p

p = 4 ¥ å n=0   (-1)n 2n+1

p = 3.14159265358979323846264338327950288419716939937510... p/2 = 1.57079632679489661923132169163975144209858469968755... p/3 = 1.04719755119659774615421446109316762806572313312503... p/4 = 0.78539816339744830961566084581987572104929234984377... p/5 = 0.62831853071795864769252867665590057683943387987502... p/6 = 0.52359877559829887307710723054658381403286156656251... 2p = 6.28318530717958647692528676655900576839433879875021... 3p = 9.42477796076937971538793014983850865259150819812531... 4p = 12.56637061435917295385057353311801153678867759750042... p2 = 9.86960440108935861883449099987615113531369940724079... p3 = 31.00627668029982017547631506710139520222528856588510... 1/p = 0.31830988618379067153776752674502872406891929148091... 2/p = 0.63661977236758134307553505349005744813783858296182... Ö   p   = 1.77245385090551602729816748334114518279754945612238... 1/ Ö   p   = 0.56418958354775628694807945156077258584405062932899... Ö   2p   = 2.50662827463100050241576528481104525300698674060993... 1/ Ö   2p   = 0.39894228040143267793994605993438186847585863116493... 3 Ö   p   = 1.46459188756152326302014252726379039173859685562793...

2  Related to e

e= ¥ å n=0   1 n! .

e = 2.71828182845904523536028747135266249775724709369995... e/2 = 1.35914091422952261768014373567633124887862354684997... 2e = 5.43656365691809047072057494270532499551449418739991... e2 = 7.38905609893065022723042746057500781318031557055184... e3 = 20.08553692318766774092852965458171789698790783855415... 1/e = 0.36787944117144232159552377016146086744581113103176... Öe = 1.64872127070012814684865078781416357165377610071014... 1/Öe = 0.60653065971263342360379953499118045344191813548718... ep = 23.14069263277926900572908636794854738026610624260021... e-p = 0.04321391826377224977441773717172801127572810981063... ep/2 = 4.81047738096535165547303566670383312639017087466453... e-p/2 = 0.20787957635076190854695561983497877003387784163176... eg = 1.78107241799019798523650410310717954916964521430343... e-g = 0.56145948356688516982414321479088078676571038692515... ee = 15.15426224147926418976043027262991190552854853685613... e-e = 0.065988035845312537076790187596846424938577048252796...

3  Related to Catalan's constant

G is defined by the series expansion

G= ¥ å n=0   (-1)n (2n+1)2 .

G = 0.91596559417721901505460351493238411077414937428167... 1/G = 1.09174406370390610145415947333389232498605012140824... G/p = 0.29156090403081878013838445646839491886406615398583... p/G = 3.42981513013245864263455323784799901211670795530093...

4  Square roots

Ö2 = 1.41421356237309504880168872420969807856967187537694... Ö3 = 1.73205080756887729352744634150587236694280525381038... Ö5 = 2.23606797749978969640917366873127623544061835961152... Ö7 = 2.64575131106459059050161575363926042571025918308245... Ö   10   = 3.16227766016837933199889354443271853371955513932521... 1/Ö2 = 0.70710678118654752440084436210484903928483593768847... 1/Ö3 = 0.57735026918962576450914878050195745564760175127012... 1/Ö5 = 0.44721359549995793928183473374625524708812367192230... f = 1.61803398874989484820458683436563811772030917980576... 3Ö2 = 1.25992104989487316476721060727822835057025146470150... 4Ö2 = 1.18920711500272106671749997056047591529297209246381... 2Ö2 = 2.66514414269022518865029724987313984827421131371465...

f = 1/2+Ö5/2 is the golden ratio.

5  Logarithms

We define the logarithm as :

log(x)= ó õ x 1   dt t .

log(2) = 0.69314718055994530941723212145817656807550013436025... log(3) = 1.09861228866810969139524523692252570464749055782274... log(5) = 1.60943791243410037460075933322618763952560135426851... log(7) = 1.94591014905531330510535274344317972963708472958186... log(10) = 2.30258509299404568401799145468436420760110148862877... 1/log(10) = 0.43429448190325182765112891891660508229439700580366...  log(2) log(3) = 0.63092975357145743709952711434276085429958564013188... log( log(2)) = -0.36651292058166432701243915823266946945426344783711... log( p) = 1.14472988584940017414342735135305871164729481291531... log æ è Ö   2p   ö ø = 0.91893853320467274178032973640561763986139747363778... log( g) = -0.54953931298164482233766176880290778833069898126306... log( f) = 0.48121182505960344749775891342436842313518433438566...

6  Gamma and Psi functions

An elementary definition is :

G(x) = ó õ ¥ 0  tx-1e-tdt,       with x > 0, y(x) =  G¢(x) G(x)

and we have the important relations

G¢(1) = -g G¢(xm) = 0.

g = 0.57721566490153286060651209008240243104215933593992... 1/g = 1.73245471460063347358302531586082968115577655226680... G(1/2) = 1.77245385090551602729816748334114518279754945612238... G(1/3) = 2.67893853470774763365569294097467764412868937795730... G(2/3) = 1.35411793942640041694528802815451378551932726605679... G(1/4) = 3.62560990822190831193068515586767200299516768288006... G(3/4) = 1.22541670246517764512909830336289052685123924810807... xm = 1.46163214496836234126265954232572132846819620400644... G(xm) = 0.88560319441088870027881590058258873320795153366990... y(1/2) = -1.96351002602142347944097633299875556719315960466043... y(1/3) = -3.13203378002080632299641907428726885415542829672041... y(2/3) = -1.31823441578658847240234081664511312187136204862767... y(1/4) = -4.22745353337626540808953014609668357736724443870824... y(3/4) = -1.08586087978647216962688676281718069317007503933313... L = 5.24411510858423962092967917978223882736550990286325...

L is the Lemniscate constant and its value is given by

L=  G2(1/4) Ö 2p .

7  Riemann Zeta function

For any s > 1

z(s)= ¥ å n=1   1 ns .

z(2) = 1.64493406684822643647241516664602518921894990120679... z(3) = 1.20205690315959428539973816151144999076498629234049... z(4) = 1.08232323371113819151600369654116790277475095191872... z(5) = 1.03692775514336992633136548645703416805708091950191... z(6) = 1.01734306198444913971451792979092052790181749003285... z(1/2) = -1.46035450880958681288949915251529801246722933101258... z(1/3) = -0.97336024835078271546888686244789657077282963174305... z(2/3) = -2.44758073623365823109099570422300521301545223575799... z(3/2) = 2.61237534868548834334856756792407163057080065240006... z(-1/2) = -0.20788622497735456601730672539704930222626853128767... 1/z(2) = 0.607927101854026628663276779258365833426152648033479... z¢(0) = -0.91893853320467274178032973640561763986139747363778... z¢(2) = -0.93754825431584375370257409456786497789786028861482...

7.1  Non trivial zeros

First roots of the equation z(s)=0 of the form

s=  1 2 +itm.

t1 = 14.13472514173469379045725198356247027078425711569924... t2 = 21.02203963877155499262847959389690277733434052490278... t3 = 25.01085758014568876321379099256282181865954967255799... t4 = 30.42487612585951321031189753058409132018156002371544... t5 = 32.93506158773918969066236896407490348881271560351703... t6 = 37.58617815882567125721776348070533282140559735083079...

You may find very complete tables on the zeros of the Riemann zeta function at [3].

8  Number theory

In this section the letter p always denotes a prime and the sums or products are only involving the set of prime numbers.

• Mertens constant M
  

  M = lim n® ¥  æ è å p £ n   1 p -log(log(n)) ö ø M = 0.26149721284764278375542683860869585905156664826119...

• Artin's constant
  

  C = Õ p ³ 2   æ è 1-  1 p(p-1) ö ø C = 0.37395581361920228805472805434641641511162924860615...

• Twin prime constant
  

  C2 = Õ p ³ 3   æ è 1-  1 (p-1)2 ö ø C2 = 0.66016181584686957392781211001455577843262336028473... 2C2 = 1.32032363169373914785562422002911155686524672056946...

• Landau-Ramanujan constant
  

  K=0.76422365358922066299069873125009232811679054139340...

References

[1]

M. Abramowitz and I. Stegun, Handbook of Mathematical Functions, Dover, New York, (1964)

[2]

D.E. Knuth, The Art of Computer Programming, Vol. II, Seminumerical Algorithms, Addison Wesley, (1998)

[3]

A. Odlyzko, Tables of zeros of the Riemann zeta function, http://www.dtc.umn.edu/~odlyzko/

[4]

S. Plouffe, Plouffe's Inverter, http://pi.lacim.uqam.ca/eng/