| L(s) = 1 | + (2.10 + 2.90i)2-s + (−2.00 − 2.23i)3-s + (−2.74 + 8.43i)4-s + (1.22 − 1.68i)5-s + (2.26 − 10.5i)6-s + (2.73 − 8.42i)7-s + (−16.6 + 5.39i)8-s + (−0.979 + 8.94i)9-s + 7.48·10-s + (−10.8 + 2.03i)11-s + (24.3 − 10.7i)12-s + (1.33 − 0.966i)13-s + (30.2 − 9.82i)14-s + (−6.22 + 0.640i)15-s + (−21.9 − 15.9i)16-s + (−7.30 + 10.0i)17-s + ⋯ |
| L(s) = 1 | + (1.05 + 1.45i)2-s + (−0.667 − 0.744i)3-s + (−0.685 + 2.10i)4-s + (0.245 − 0.337i)5-s + (0.376 − 1.75i)6-s + (0.391 − 1.20i)7-s + (−2.07 + 0.674i)8-s + (−0.108 + 0.994i)9-s + 0.748·10-s + (−0.982 + 0.184i)11-s + (2.02 − 0.897i)12-s + (0.102 − 0.0743i)13-s + (2.15 − 0.701i)14-s + (−0.415 + 0.0427i)15-s + (−1.37 − 0.998i)16-s + (−0.429 + 0.591i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.355 - 0.934i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.09903 + 0.757545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.09903 + 0.757545i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (2.00 + 2.23i)T \) |
| 11 | \( 1 + (10.8 - 2.03i)T \) |
| good | 2 | \( 1 + (-2.10 - 2.90i)T + (-1.23 + 3.80i)T^{2} \) |
| 5 | \( 1 + (-1.22 + 1.68i)T + (-7.72 - 23.7i)T^{2} \) |
| 7 | \( 1 + (-2.73 + 8.42i)T + (-39.6 - 28.8i)T^{2} \) |
| 13 | \( 1 + (-1.33 + 0.966i)T + (52.2 - 160. i)T^{2} \) |
| 17 | \( 1 + (7.30 - 10.0i)T + (-89.3 - 274. i)T^{2} \) |
| 19 | \( 1 + (-3.26 - 10.0i)T + (-292. + 212. i)T^{2} \) |
| 23 | \( 1 + 20.3iT - 529T^{2} \) |
| 29 | \( 1 + (11.0 + 3.58i)T + (680. + 494. i)T^{2} \) |
| 31 | \( 1 + (-18.9 + 13.7i)T + (296. - 913. i)T^{2} \) |
| 37 | \( 1 + (-2.23 + 6.89i)T + (-1.10e3 - 804. i)T^{2} \) |
| 41 | \( 1 + (-36.9 + 12.0i)T + (1.35e3 - 988. i)T^{2} \) |
| 43 | \( 1 + 15.8T + 1.84e3T^{2} \) |
| 47 | \( 1 + (-43.0 + 14.0i)T + (1.78e3 - 1.29e3i)T^{2} \) |
| 53 | \( 1 + (-23.6 - 32.6i)T + (-868. + 2.67e3i)T^{2} \) |
| 59 | \( 1 + (107. + 34.9i)T + (2.81e3 + 2.04e3i)T^{2} \) |
| 61 | \( 1 + (62.7 + 45.5i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 - 62.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + (-6.00 + 8.26i)T + (-1.55e3 - 4.79e3i)T^{2} \) |
| 73 | \( 1 + (23.0 - 70.9i)T + (-4.31e3 - 3.13e3i)T^{2} \) |
| 79 | \( 1 + (69.8 - 50.7i)T + (1.92e3 - 5.93e3i)T^{2} \) |
| 83 | \( 1 + (6.94 - 9.55i)T + (-2.12e3 - 6.55e3i)T^{2} \) |
| 89 | \( 1 + 74.5iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-62.4 + 45.3i)T + (2.90e3 - 8.94e3i)T^{2} \) |
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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
−16.83859912566041522466569644996, −15.60055904491388120107658169167, −14.16580993130222377030202305089, −13.30990382921610648882621533058, −12.54348949575030349485121667082, −10.75422782969723137616663603747, −8.092891232405774720189514998888, −7.18492034778396985841818290100, −5.82733135402337444858533630108, −4.53125510935500220645658031454,
2.79130138692981389927296819600, 4.79320695853155673146892708004, 5.82450730489962965267323670818, 9.232702530543916121450087408719, 10.49766152420418499505124326589, 11.41007152334896715564050240576, 12.29213522403438141662776722621, 13.63635549964826958116565059738, 14.94169556764316542901625878474, 15.81420089775981826199523461190
