L(s) = 1 | + (−0.994 − 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (−0.743 − 0.669i)6-s + (−0.406 + 0.913i)7-s + (−0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (0.743 − 0.669i)10-s − i·11-s + (0.669 + 0.743i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (−0.207 + 0.978i)17-s + ⋯ |
L(s) = 1 | + (−0.994 − 0.104i)2-s + (0.809 + 0.587i)3-s + (0.978 + 0.207i)4-s + (−0.669 + 0.743i)5-s + (−0.743 − 0.669i)6-s + (−0.406 + 0.913i)7-s + (−0.951 − 0.309i)8-s + (0.309 + 0.951i)9-s + (0.743 − 0.669i)10-s − i·11-s + (0.669 + 0.743i)12-s + (−0.5 − 0.866i)13-s + (0.5 − 0.866i)14-s + (−0.978 + 0.207i)15-s + (0.913 + 0.406i)16-s + (−0.207 + 0.978i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 61 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.888 + 0.458i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1678625926 + 0.6922489461i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1678625926 + 0.6922489461i\) |
\(L(1)\) |
\(\approx\) |
\(0.6013453677 + 0.3310676054i\) |
\(L(1)\) |
\(\approx\) |
\(0.6013453677 + 0.3310676054i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 61 | \( 1 \) |
good | 2 | \( 1 + (-0.994 - 0.104i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 5 | \( 1 + (-0.669 + 0.743i)T \) |
| 7 | \( 1 + (-0.406 + 0.913i)T \) |
| 11 | \( 1 - iT \) |
| 13 | \( 1 + (-0.5 - 0.866i)T \) |
| 17 | \( 1 + (-0.207 + 0.978i)T \) |
| 19 | \( 1 + (-0.913 + 0.406i)T \) |
| 23 | \( 1 + (-0.951 + 0.309i)T \) |
| 29 | \( 1 + (0.866 + 0.5i)T \) |
| 31 | \( 1 + (-0.994 + 0.104i)T \) |
| 37 | \( 1 + (0.587 + 0.809i)T \) |
| 41 | \( 1 + (0.809 - 0.587i)T \) |
| 43 | \( 1 + (0.207 + 0.978i)T \) |
| 47 | \( 1 + (-0.5 + 0.866i)T \) |
| 53 | \( 1 + (0.951 + 0.309i)T \) |
| 59 | \( 1 + (0.994 + 0.104i)T \) |
| 67 | \( 1 + (-0.743 - 0.669i)T \) |
| 71 | \( 1 + (0.743 - 0.669i)T \) |
| 73 | \( 1 + (0.669 + 0.743i)T \) |
| 79 | \( 1 + (-0.207 - 0.978i)T \) |
| 83 | \( 1 + (-0.104 + 0.994i)T \) |
| 89 | \( 1 + (-0.587 + 0.809i)T \) |
| 97 | \( 1 + (0.104 + 0.994i)T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−31.768949421357819769878229240998, −30.55310881804372898245658468182, −29.47767230369169982963652188967, −28.4355568015279058763328056204, −27.13805405877076652428594954119, −26.2617539111583142059614013243, −25.24906258749524570280054088773, −24.1538893359475474423896262849, −23.32581063885687394578627955272, −20.93486751330870246110209134417, −19.92946597754662277771718136809, −19.53251336522678773258731063428, −18.1130369998762200050688218742, −16.87569897189301775271909387043, −15.76886179192908279800346672744, −14.45050144031036000287536234499, −12.85432051489801631757687414519, −11.72838229088631053910929948997, −9.89839359673379031911507492419, −8.8878894966583273145608122305, −7.58979411002091300995928420839, −6.83839559084844626016273694138, −4.174715486060199471189175111825, −2.16696435049853041419261342234, −0.450071452386221704518884042223,
2.48297072756017107051088357558, 3.54582748337481310251690684322, 6.12129061423707538050460884692, 7.87031422074853576634555235873, 8.65673006520687751336006596396, 10.07423465874375133411094622218, 11.034565466676147522868117956318, 12.564656749873659358935903210379, 14.64326673238376639937286672788, 15.46382805548580377589444376429, 16.39818728760665859739938878932, 18.154386865888844197402628599904, 19.28485066426057333315484256784, 19.73475195777284020872268445395, 21.36755928926727590979900017277, 22.15537451549947695228235754497, 24.14219629459299861553122038173, 25.40632667781018573425764070797, 26.121847324840459949123689489499, 27.24424751315095372405730382491, 27.801074747315083832050194429030, 29.37386764732282019773475132608, 30.47797191487123900918984370392, 31.58794764094389402716680599705, 32.6793104694513078102789807466