L(s) = 1 | + (1.99 + 0.726i)2-s + (1.92 + 1.61i)4-s + (−0.359 + 2.03i)5-s + (3.71 − 3.11i)7-s + (0.547 + 0.949i)8-s + (−2.20 + 3.81i)10-s + (0.720 + 4.08i)11-s + (1.14 − 0.415i)13-s + (9.68 − 3.52i)14-s + (−0.469 − 2.66i)16-s + (−1.18 + 2.04i)17-s + (0.919 + 1.59i)19-s + (−3.99 + 3.34i)20-s + (−1.53 + 8.68i)22-s + (−3.29 − 2.76i)23-s + ⋯ |
L(s) = 1 | + (1.41 + 0.513i)2-s + (0.963 + 0.808i)4-s + (−0.160 + 0.912i)5-s + (1.40 − 1.17i)7-s + (0.193 + 0.335i)8-s + (−0.695 + 1.20i)10-s + (0.217 + 1.23i)11-s + (0.316 − 0.115i)13-s + (2.58 − 0.941i)14-s + (−0.117 − 0.665i)16-s + (−0.286 + 0.496i)17-s + (0.210 + 0.365i)19-s + (−0.892 + 0.748i)20-s + (−0.326 + 1.85i)22-s + (−0.687 − 0.576i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.597 - 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.13868 + 1.57630i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.13868 + 1.57630i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.99 - 0.726i)T + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (0.359 - 2.03i)T + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (-3.71 + 3.11i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-0.720 - 4.08i)T + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-1.14 + 0.415i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (1.18 - 2.04i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.919 - 1.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.29 + 2.76i)T + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (2.80 + 1.01i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-1.12 - 0.947i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (4.48 - 7.76i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.12 - 0.773i)T + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (0.952 + 5.40i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-5.50 + 4.61i)T + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 6.32T + 53T^{2} \) |
| 59 | \( 1 + (-0.0455 + 0.258i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (3.41 - 2.86i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (3.88 - 1.41i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (1.54 - 2.67i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (6.38 + 11.0i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4.27 + 1.55i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (-7.94 - 2.89i)T + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (8.48 + 14.6i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (0.887 + 5.03i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.61824638687717921313435547341, −10.04928360137827523066991451178, −8.444041649033216710175235404231, −7.39936768491866321137372128691, −7.05302324574719254755246654953, −6.05805260858870319692215412441, −4.83585441909194593273983548962, −4.28774885263620436888773361740, −3.38756370988633368751773359421, −1.82881475737310755686955108373,
1.50562230357598342551860893109, 2.67423941281932973239710249762, 3.90237060006861297329408920039, 4.87083457646151190641160781975, 5.42577146466280926517624967951, 6.16186965778636064913654334708, 7.81743924003950401067483465070, 8.678922500157223953605774763936, 9.129781361553104965330947278269, 10.89089635642108474162902505128