| L(s) = 1 | + (−0.807 − 0.677i)2-s + (−0.154 − 0.874i)4-s + (−1.64 − 0.597i)5-s + (0.427 − 2.42i)7-s + (−1.52 + 2.63i)8-s + (0.920 + 1.59i)10-s + (−1.17 + 0.428i)11-s + (−3.48 + 2.92i)13-s + (−1.98 + 1.66i)14-s + (1.34 − 0.490i)16-s + (−3.32 − 5.75i)17-s + (−0.124 + 0.215i)19-s + (−0.269 + 1.52i)20-s + (1.24 + 0.452i)22-s + (−0.146 − 0.829i)23-s + ⋯ |
| L(s) = 1 | + (−0.571 − 0.479i)2-s + (−0.0771 − 0.437i)4-s + (−0.733 − 0.267i)5-s + (0.161 − 0.915i)7-s + (−0.538 + 0.932i)8-s + (0.291 + 0.504i)10-s + (−0.355 + 0.129i)11-s + (−0.966 + 0.810i)13-s + (−0.530 + 0.445i)14-s + (0.336 − 0.122i)16-s + (−0.806 − 1.39i)17-s + (−0.0285 + 0.0495i)19-s + (−0.0602 + 0.341i)20-s + (0.264 + 0.0964i)22-s + (−0.0304 − 0.172i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.00522i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.00522i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.00111184 + 0.425372i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00111184 + 0.425372i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 + (0.807 + 0.677i)T + (0.347 + 1.96i)T^{2} \) |
| 5 | \( 1 + (1.64 + 0.597i)T + (3.83 + 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.427 + 2.42i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (1.17 - 0.428i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.48 - 2.92i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (3.32 + 5.75i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (0.124 - 0.215i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.146 + 0.829i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.392 - 0.329i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.142 + 0.807i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (1.30 + 2.25i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.24 + 5.24i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.06 - 1.47i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-0.920 + 5.22i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 10.4T + 53T^{2} \) |
| 59 | \( 1 + (2.82 + 1.02i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-0.500 + 2.83i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-7.72 + 6.48i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-0.0447 - 0.0774i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.66 + 4.60i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-3.65 - 3.07i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (6.15 + 5.16i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.35 + 5.80i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (5.15 - 1.87i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.49961719332017066049355667082, −10.66882722871916613142003775159, −9.745114285918060691198207418297, −8.917728892556319530391977762905, −7.73921620366257885919960427504, −6.84689161951212535844186793285, −5.14813316714560882566807626137, −4.21694771623556521885723953497, −2.31738795968408789623287814477, −0.39417986311437833666785669476,
2.71944206941413869255470033158, 4.05693314626167731159293054680, 5.59099031685427733921821361783, 6.84333565292575612884774367423, 7.894625240282956196225793019366, 8.414876898485462181547380085139, 9.475832111152486374846310810547, 10.61481663500938165054813341290, 11.74186357015763147637479405574, 12.49413763747325403968100203294
