L(s) = 1 | + (0.962 + 0.269i)2-s + (0.460 − 0.887i)3-s + (0.854 + 0.519i)4-s + (−0.775 + 0.631i)5-s + (0.682 − 0.730i)6-s + (−0.990 − 0.136i)7-s + (0.682 + 0.730i)8-s + (−0.576 − 0.816i)9-s + (−0.917 + 0.398i)10-s + (−0.0682 − 0.997i)11-s + (0.854 − 0.519i)12-s + (−0.334 + 0.942i)13-s + (−0.917 − 0.398i)14-s + (0.203 + 0.979i)15-s + (0.460 + 0.887i)16-s + (−0.0682 + 0.997i)17-s + ⋯ |
L(s) = 1 | + (0.962 + 0.269i)2-s + (0.460 − 0.887i)3-s + (0.854 + 0.519i)4-s + (−0.775 + 0.631i)5-s + (0.682 − 0.730i)6-s + (−0.990 − 0.136i)7-s + (0.682 + 0.730i)8-s + (−0.576 − 0.816i)9-s + (−0.917 + 0.398i)10-s + (−0.0682 − 0.997i)11-s + (0.854 − 0.519i)12-s + (−0.334 + 0.942i)13-s + (−0.917 − 0.398i)14-s + (0.203 + 0.979i)15-s + (0.460 + 0.887i)16-s + (−0.0682 + 0.997i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.998 - 0.0450i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.332652606 + 0.03004588099i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.332652606 + 0.03004588099i\) |
\(L(1)\) |
\(\approx\) |
\(1.508829270 + 0.01651739324i\) |
\(L(1)\) |
\(\approx\) |
\(1.508829270 + 0.01651739324i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 \) |
good | 2 | \( 1 + (0.962 + 0.269i)T \) |
| 3 | \( 1 + (0.460 - 0.887i)T \) |
| 5 | \( 1 + (-0.775 + 0.631i)T \) |
| 7 | \( 1 + (-0.990 - 0.136i)T \) |
| 11 | \( 1 + (-0.0682 - 0.997i)T \) |
| 13 | \( 1 + (-0.334 + 0.942i)T \) |
| 17 | \( 1 + (-0.0682 + 0.997i)T \) |
| 19 | \( 1 + (-0.775 - 0.631i)T \) |
| 23 | \( 1 + (0.962 - 0.269i)T \) |
| 29 | \( 1 + (-0.334 - 0.942i)T \) |
| 31 | \( 1 + (0.460 + 0.887i)T \) |
| 37 | \( 1 + (-0.917 + 0.398i)T \) |
| 41 | \( 1 + (0.682 - 0.730i)T \) |
| 43 | \( 1 + (0.854 + 0.519i)T \) |
| 53 | \( 1 + (0.682 - 0.730i)T \) |
| 59 | \( 1 + (0.854 - 0.519i)T \) |
| 61 | \( 1 + (-0.917 - 0.398i)T \) |
| 67 | \( 1 + (-0.990 + 0.136i)T \) |
| 71 | \( 1 + (0.962 - 0.269i)T \) |
| 73 | \( 1 + (-0.576 + 0.816i)T \) |
| 79 | \( 1 + (0.203 + 0.979i)T \) |
| 83 | \( 1 + (-0.0682 - 0.997i)T \) |
| 89 | \( 1 + (-0.775 + 0.631i)T \) |
| 97 | \( 1 + (0.460 - 0.887i)T \) |
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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
−33.62738519590339069994778171519, −32.634311734363276741199722735577, −31.723798025332823509390065426797, −31.19606686141893109848988620392, −29.57626743920838919788031004563, −28.27533623632207343510955140176, −27.396207777761531001170396597655, −25.67653811882036916707541503147, −24.801566860244628917066566510224, −23.02931972222932863538701737117, −22.531364628851368525446571963, −20.947398829160435009805207230435, −20.151078233549935057544656299049, −19.26710005989781703626016338682, −16.66249672212835606397890693825, −15.627532004008247588170059963, −14.898530798193868926279632425325, −13.18328668058326132382760955070, −12.184278402602451252589133947933, −10.57512152753373067584237222065, −9.314750184413034882276520445227, −7.415154299485583695955339488146, −5.33364335886817658482282018786, −4.13510287445514101048688333155, −2.84522823346548694262016337874,
2.68281503458342409014602386626, 3.85522928736977886773070235763, 6.28921819437613647477591305682, 7.07001732369226753931987586746, 8.532243710653705659140507477553, 10.990117059566438455915844825172, 12.26343277198395745342144932314, 13.35227581968450979355176943331, 14.46789913255941297455664837696, 15.63388433280456610300882893762, 17.02581740758096439948964756791, 19.17443881921856392001111647151, 19.44486820136920854181898452990, 21.25802428581839521456980288727, 22.60267935322422441824366184767, 23.589149103799783921554801891620, 24.40858462817196605523425138644, 25.8944950377307120211986898789, 26.492987887284124325707376838414, 28.85720717554296284420273018574, 29.79560905694960031767692442395, 30.75236812832279165606709622754, 31.67955901502825625523980113235, 32.55295762564614517198420922203, 34.331401480298206522285034110108