L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.173 + 0.984i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + i·6-s + (0.766 + 0.642i)7-s + (0.866 − 0.5i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.342 − 0.939i)13-s + (0.866 + 0.5i)14-s + (−0.642 − 0.766i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + ⋯ |
L(s) = 1 | + (0.984 − 0.173i)2-s + (−0.173 + 0.984i)3-s + (0.939 − 0.342i)4-s + (−0.642 + 0.766i)5-s + i·6-s + (0.766 + 0.642i)7-s + (0.866 − 0.5i)8-s + (−0.939 − 0.342i)9-s + (−0.5 + 0.866i)10-s + (0.5 + 0.866i)11-s + (0.173 + 0.984i)12-s + (−0.342 − 0.939i)13-s + (0.866 + 0.5i)14-s + (−0.642 − 0.766i)15-s + (0.766 − 0.642i)16-s + (0.342 − 0.939i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 37 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.512 + 0.858i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.029642993 + 1.152418744i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.029642993 + 1.152418744i\) |
\(L(1)\) |
\(\approx\) |
\(1.653255768 + 0.5516078989i\) |
\(L(1)\) |
\(\approx\) |
\(1.653255768 + 0.5516078989i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
good | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 5 | \( 1 + (-0.642 + 0.766i)T \) |
| 7 | \( 1 + (0.766 + 0.642i)T \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
| 13 | \( 1 + (-0.342 - 0.939i)T \) |
| 17 | \( 1 + (0.342 - 0.939i)T \) |
| 19 | \( 1 + (0.984 + 0.173i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.866 + 0.5i)T \) |
| 31 | \( 1 - iT \) |
| 41 | \( 1 + (0.939 - 0.342i)T \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.766 - 0.642i)T \) |
| 59 | \( 1 + (0.642 + 0.766i)T \) |
| 61 | \( 1 + (0.342 + 0.939i)T \) |
| 67 | \( 1 + (-0.766 - 0.642i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 - T \) |
| 79 | \( 1 + (-0.642 + 0.766i)T \) |
| 83 | \( 1 + (-0.939 - 0.342i)T \) |
| 89 | \( 1 + (-0.642 - 0.766i)T \) |
| 97 | \( 1 + (-0.866 - 0.5i)T \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−34.98346255913779239230214836787, −33.975881811278088004903177203154, −32.56971375497625436208076869893, −31.387207171522633674608902175, −30.49289700059897324324187130127, −29.48118741899830929591542250185, −28.21880505494170342866723872618, −26.44954810403971653616269536635, −24.672714586227949795279618810907, −24.07364326019676342136346819837, −23.327743534559870533724622068408, −21.7146331793357576380902012780, −20.26166330137894580095685872659, −19.29617169491723829261100081393, −17.2562987088248078655300158787, −16.31796750974118115881309770767, −14.43271098264807754216196488123, −13.472970761409397868514957034955, −12.05835175025640528013488945652, −11.30886918557010907795247944203, −8.28910429531704823716689154774, −7.17943611923155048660068156458, −5.51917955363491889972675771657, −3.904718325418339776650738173407, −1.464309442509693827030654868096,
2.81510137788842592327727230104, 4.320472306279818767382239396817, 5.6387110901833294827214847638, 7.57307444164045909047947241601, 9.893761430538187515638509401235, 11.27300951889973917535260613660, 12.109676195511476677689243333766, 14.42247372080694615253223712934, 15.02499065131812529740285527919, 16.130916600865034769333087348662, 18.041822795539283489093770036115, 19.93159114567857326318758135216, 20.86155622401941499786307783068, 22.4337943871783468752265321123, 22.61350622655063081163567690176, 24.355143792537678502559069791500, 25.732087226723672724700472327219, 27.32681985525169004569146119512, 28.14411509315394186251331782393, 29.81784456905107429410710071583, 30.97933580011500989973550983735, 31.79727811854931525651258146105, 33.16197773794718088470250545720, 34.0319207799969995414873483877, 34.90401234442120974284536167365