| L(s) = 1 | + (1.81 + 1.05i)2-s + (4.83 + 1.90i)3-s + (−1.79 − 3.10i)4-s + (3.96 + 1.06i)5-s + (6.80 + 8.54i)6-s + (15.5 − 4.17i)7-s − 24.3i·8-s + (19.7 + 18.4i)9-s + (6.10 + 6.10i)10-s + (0.247 − 0.0662i)11-s + (−2.75 − 18.4i)12-s + (4.25 + 7.36i)13-s + (32.7 + 8.77i)14-s + (17.1 + 12.6i)15-s + (11.2 − 19.4i)16-s + (62.5 + 31.6i)17-s + ⋯ |
| L(s) = 1 | + (0.643 + 0.371i)2-s + (0.930 + 0.366i)3-s + (−0.224 − 0.388i)4-s + (0.354 + 0.0950i)5-s + (0.462 + 0.581i)6-s + (0.841 − 0.225i)7-s − 1.07i·8-s + (0.731 + 0.681i)9-s + (0.192 + 0.192i)10-s + (0.00677 − 0.00181i)11-s + (−0.0663 − 0.443i)12-s + (0.0906 + 0.157i)13-s + (0.625 + 0.167i)14-s + (0.295 + 0.218i)15-s + (0.175 − 0.304i)16-s + (0.892 + 0.451i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 153 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.950 - 0.311i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(3.26073 + 0.521245i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.26073 + 0.521245i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.83 - 1.90i)T \) |
| 17 | \( 1 + (-62.5 - 31.6i)T \) |
| good | 2 | \( 1 + (-1.81 - 1.05i)T + (4 + 6.92i)T^{2} \) |
| 5 | \( 1 + (-3.96 - 1.06i)T + (108. + 62.5i)T^{2} \) |
| 7 | \( 1 + (-15.5 + 4.17i)T + (297. - 171.5i)T^{2} \) |
| 11 | \( 1 + (-0.247 + 0.0662i)T + (1.15e3 - 665.5i)T^{2} \) |
| 13 | \( 1 + (-4.25 - 7.36i)T + (-1.09e3 + 1.90e3i)T^{2} \) |
| 19 | \( 1 + 48.7iT - 6.85e3T^{2} \) |
| 23 | \( 1 + (41.7 - 155. i)T + (-1.05e4 - 6.08e3i)T^{2} \) |
| 29 | \( 1 + (37.4 + 139. i)T + (-2.11e4 + 1.21e4i)T^{2} \) |
| 31 | \( 1 + (13.1 + 3.51i)T + (2.57e4 + 1.48e4i)T^{2} \) |
| 37 | \( 1 + (141. - 141. i)T - 5.06e4iT^{2} \) |
| 41 | \( 1 + (32.1 - 120. i)T + (-5.96e4 - 3.44e4i)T^{2} \) |
| 43 | \( 1 + (59.5 + 34.3i)T + (3.97e4 + 6.88e4i)T^{2} \) |
| 47 | \( 1 + (-114. + 198. i)T + (-5.19e4 - 8.99e4i)T^{2} \) |
| 53 | \( 1 + 548. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + (-104. + 60.2i)T + (1.02e5 - 1.77e5i)T^{2} \) |
| 61 | \( 1 + (178. - 47.7i)T + (1.96e5 - 1.13e5i)T^{2} \) |
| 67 | \( 1 + (259. + 449. i)T + (-1.50e5 + 2.60e5i)T^{2} \) |
| 71 | \( 1 + (176. - 176. i)T - 3.57e5iT^{2} \) |
| 73 | \( 1 + (614. - 614. i)T - 3.89e5iT^{2} \) |
| 79 | \( 1 + (809. - 216. i)T + (4.26e5 - 2.46e5i)T^{2} \) |
| 83 | \( 1 + (215. + 124. i)T + (2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 - 356.T + 7.04e5T^{2} \) |
| 97 | \( 1 + (91.2 + 340. i)T + (-7.90e5 + 4.56e5i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.09092405827955704790398224702, −11.58762452880075272265822605280, −10.23191014928842823405863480772, −9.619013052411797591081968713992, −8.352537490030144545679930157210, −7.28895009252893934263625506459, −5.80079637887393645348686623351, −4.70130581031580875972342749222, −3.60160694863108912364948352779, −1.67060236747642813115214233786,
1.79235299401993800965859739527, 3.08935961569322946728838488555, 4.35094546749934412504853772877, 5.65305278288237753418006574254, 7.43871756734739614668230147015, 8.291419326795649154802191892643, 9.154446129443492625076869657892, 10.51173597377507638214866493352, 11.90314543591948032407418028941, 12.51508792324805461708000413644
