| L(s) = 1 | + (0.839 − 0.839i)3-s + (−99.3 − 99.3i)7-s + 241. i·9-s − 637. i·11-s + (640. + 640. i)13-s + (−648. + 648. i)17-s + 2.50e3·19-s − 166.·21-s + (−2.80e3 + 2.80e3i)23-s + (406. + 406. i)27-s − 4.95e3i·29-s − 1.96e3i·31-s + (−535. − 535. i)33-s + (1.89e3 − 1.89e3i)37-s + 1.07e3·39-s + ⋯ |
| L(s) = 1 | + (0.0538 − 0.0538i)3-s + (−0.766 − 0.766i)7-s + 0.994i·9-s − 1.58i·11-s + (1.05 + 1.05i)13-s + (−0.544 + 0.544i)17-s + 1.59·19-s − 0.0825·21-s + (−1.10 + 1.10i)23-s + (0.107 + 0.107i)27-s − 1.09i·29-s − 0.366i·31-s + (−0.0856 − 0.0856i)33-s + (0.227 − 0.227i)37-s + 0.113·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0299 + 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.0299 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.493391814\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.493391814\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + (-0.839 + 0.839i)T - 243iT^{2} \) |
| 7 | \( 1 + (99.3 + 99.3i)T + 1.68e4iT^{2} \) |
| 11 | \( 1 + 637. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-640. - 640. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (648. - 648. i)T - 1.41e6iT^{2} \) |
| 19 | \( 1 - 2.50e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (2.80e3 - 2.80e3i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 4.95e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.96e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.89e3 + 1.89e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 + 5.82e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + (-1.06e4 + 1.06e4i)T - 1.47e8iT^{2} \) |
| 47 | \( 1 + (8.30e3 + 8.30e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (7.43e3 + 7.43e3i)T + 4.18e8iT^{2} \) |
| 59 | \( 1 + 1.67e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 2.37e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (4.15e3 + 4.15e3i)T + 1.35e9iT^{2} \) |
| 71 | \( 1 - 1.22e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (3.60e4 + 3.60e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 - 6.43e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + (-6.28e4 + 6.28e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 2.44e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-8.96e4 + 8.96e4i)T - 8.58e9iT^{2} \) |
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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
−10.28691779013195399764924532875, −9.340418220461979567559079271052, −8.345530748707005957005415818115, −7.50246963443851137365143734758, −6.36205756345806238842793073721, −5.59504667040324535924016159918, −4.06493178670315770985440805520, −3.31643797998057942634889308814, −1.74663653428394800200385680805, −0.41585492723339501593748943006,
1.06264290682430618726279382730, 2.63134782711892111959595276472, 3.57216727237300155393173215497, 4.88956309404279606102202808747, 6.03028953494149344886368587727, 6.80365208317768308540435340710, 7.923722474477498205709059442478, 9.100051846279685927794335136380, 9.621530405420484047120365955914, 10.52499368773237289112537953558
