L(s) = 1 | + (0.986 − 0.161i)3-s + (0.965 − 0.259i)5-s + (−0.817 + 0.575i)7-s + (0.947 − 0.319i)9-s + (−0.0437 − 0.999i)11-s + (−0.384 + 0.923i)13-s + (0.910 − 0.412i)15-s + (0.877 − 0.479i)17-s + (0.858 + 0.512i)19-s + (−0.713 + 0.700i)21-s + (−0.418 − 0.908i)23-s + (0.864 − 0.501i)25-s + (0.883 − 0.468i)27-s + (−0.852 − 0.523i)29-s + (0.289 + 0.957i)31-s + ⋯ |
L(s) = 1 | + (0.986 − 0.161i)3-s + (0.965 − 0.259i)5-s + (−0.817 + 0.575i)7-s + (0.947 − 0.319i)9-s + (−0.0437 − 0.999i)11-s + (−0.384 + 0.923i)13-s + (0.910 − 0.412i)15-s + (0.877 − 0.479i)17-s + (0.858 + 0.512i)19-s + (−0.713 + 0.700i)21-s + (−0.418 − 0.908i)23-s + (0.864 − 0.501i)25-s + (0.883 − 0.468i)27-s + (−0.852 − 0.523i)29-s + (0.289 + 0.957i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4024 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.644 - 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.681439842 - 1.246434602i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.681439842 - 1.246434602i\) |
\(L(1)\) |
\(\approx\) |
\(1.653372195 - 0.2705101542i\) |
\(L(1)\) |
\(\approx\) |
\(1.653372195 - 0.2705101542i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 503 | \( 1 \) |
good | 3 | \( 1 + (0.986 - 0.161i)T \) |
| 5 | \( 1 + (0.965 - 0.259i)T \) |
| 7 | \( 1 + (-0.817 + 0.575i)T \) |
| 11 | \( 1 + (-0.0437 - 0.999i)T \) |
| 13 | \( 1 + (-0.384 + 0.923i)T \) |
| 17 | \( 1 + (0.877 - 0.479i)T \) |
| 19 | \( 1 + (0.858 + 0.512i)T \) |
| 23 | \( 1 + (-0.418 - 0.908i)T \) |
| 29 | \( 1 + (-0.852 - 0.523i)T \) |
| 31 | \( 1 + (0.289 + 0.957i)T \) |
| 37 | \( 1 + (-0.992 + 0.124i)T \) |
| 41 | \( 1 + (-0.485 - 0.874i)T \) |
| 43 | \( 1 + (-0.0687 - 0.997i)T \) |
| 47 | \( 1 + (-0.253 - 0.967i)T \) |
| 53 | \( 1 + (0.858 - 0.512i)T \) |
| 59 | \( 1 + (0.943 + 0.331i)T \) |
| 61 | \( 1 + (0.205 - 0.978i)T \) |
| 67 | \( 1 + (-0.910 - 0.412i)T \) |
| 71 | \( 1 + (-0.722 - 0.691i)T \) |
| 73 | \( 1 + (0.0187 + 0.999i)T \) |
| 79 | \( 1 + (0.920 + 0.389i)T \) |
| 83 | \( 1 + (0.915 - 0.401i)T \) |
| 89 | \( 1 + (0.429 + 0.902i)T \) |
| 97 | \( 1 + (0.968 + 0.247i)T \) |
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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
−18.57463083768481305114657245341, −17.86711766917661340134309417675, −17.31715205611433012170422920272, −16.48170832564891537904953365959, −15.78339288596759863120484370454, −14.92480189726072690541667641209, −14.62852396788138103631293012507, −13.68034634970618504025518619494, −13.20430551932726401955792589099, −12.766777880979065511705084675383, −11.821387340840916696989726022376, −10.52742156636282699417492836842, −10.16542095404262180410836949841, −9.56403723005272433731996928307, −9.18936837905831535803965498922, −7.91595559265753578668438403448, −7.45958371493350826142172805653, −6.81317744369066528334884154356, −5.82986640084071372330778386769, −5.1286904090410444903095269311, −4.16839529748661601691832181563, −3.26608281042459482033152638051, −2.86395567597970760783072306032, −1.88526450800533310871107494609, −1.140565144581060595141760464703,
0.72495754309366864904191917822, 1.832070601628137230729039329843, 2.37244163488848858255385678880, 3.26850856579991328848993610359, 3.76058024561253559695721768328, 5.065333990366660705749101095019, 5.624914038178980005788646040096, 6.53173776847980280276591072994, 7.05255949742632670832224309394, 8.09603733807432570763990364691, 8.84786294970389016935107287012, 9.19945836586442346890677821648, 10.01564904900886548262088633454, 10.38228920813357155151321233052, 11.902374087422601497180435740576, 12.17118887337628093936483635557, 13.10345799353904672202149669481, 13.76570111503396869566807866418, 14.07673568398959484779624001929, 14.793435547723565952831746190707, 15.74181801001068782938465225373, 16.40591351007705909193921231581, 16.75120790472932379735379020388, 17.919268967089125245341850151721, 18.6294165533537685192721790713