| L(s) = 1 | + (−0.841 − 0.540i)2-s + (0.415 + 0.909i)4-s + (−1.18 − 0.348i)5-s + (0.968 − 1.11i)7-s + (0.142 − 0.989i)8-s + (0.809 + 0.934i)10-s + (0.745 − 0.479i)11-s + (−0.0440 − 0.0508i)13-s + (−1.41 + 0.416i)14-s + (−0.654 + 0.755i)16-s + (1.78 − 3.90i)17-s + (−1.93 − 4.24i)19-s + (−0.175 − 1.22i)20-s − 0.886·22-s + (4.72 + 0.847i)23-s + ⋯ |
| L(s) = 1 | + (−0.594 − 0.382i)2-s + (0.207 + 0.454i)4-s + (−0.530 − 0.155i)5-s + (0.366 − 0.422i)7-s + (0.0503 − 0.349i)8-s + (0.256 + 0.295i)10-s + (0.224 − 0.144i)11-s + (−0.0122 − 0.0141i)13-s + (−0.379 + 0.111i)14-s + (−0.163 + 0.188i)16-s + (0.432 − 0.946i)17-s + (−0.444 − 0.974i)19-s + (−0.0393 − 0.273i)20-s − 0.189·22-s + (0.984 + 0.176i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00569 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.00569 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.633875 - 0.630274i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.633875 - 0.630274i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 + 0.540i)T \) |
| 3 | \( 1 \) |
| 23 | \( 1 + (-4.72 - 0.847i)T \) |
| good | 5 | \( 1 + (1.18 + 0.348i)T + (4.20 + 2.70i)T^{2} \) |
| 7 | \( 1 + (-0.968 + 1.11i)T + (-0.996 - 6.92i)T^{2} \) |
| 11 | \( 1 + (-0.745 + 0.479i)T + (4.56 - 10.0i)T^{2} \) |
| 13 | \( 1 + (0.0440 + 0.0508i)T + (-1.85 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.78 + 3.90i)T + (-11.1 - 12.8i)T^{2} \) |
| 19 | \( 1 + (1.93 + 4.24i)T + (-12.4 + 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.18 + 6.98i)T + (-18.9 - 21.9i)T^{2} \) |
| 31 | \( 1 + (-0.226 + 1.57i)T + (-29.7 - 8.73i)T^{2} \) |
| 37 | \( 1 + (-5.01 + 1.47i)T + (31.1 - 20.0i)T^{2} \) |
| 41 | \( 1 + (0.130 + 0.0382i)T + (34.4 + 22.1i)T^{2} \) |
| 43 | \( 1 + (0.936 + 6.51i)T + (-41.2 + 12.1i)T^{2} \) |
| 47 | \( 1 + 8.31T + 47T^{2} \) |
| 53 | \( 1 + (1.96 - 2.26i)T + (-7.54 - 52.4i)T^{2} \) |
| 59 | \( 1 + (4.47 + 5.16i)T + (-8.39 + 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.654 + 4.55i)T + (-58.5 - 17.1i)T^{2} \) |
| 67 | \( 1 + (-10.4 - 6.70i)T + (27.8 + 60.9i)T^{2} \) |
| 71 | \( 1 + (-6.15 - 3.95i)T + (29.4 + 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.48 - 12.0i)T + (-47.8 + 55.1i)T^{2} \) |
| 79 | \( 1 + (5.16 + 5.96i)T + (-11.2 + 78.1i)T^{2} \) |
| 83 | \( 1 + (8.06 - 2.36i)T + (69.8 - 44.8i)T^{2} \) |
| 89 | \( 1 + (-1.66 - 11.6i)T + (-85.3 + 25.0i)T^{2} \) |
| 97 | \( 1 + (-16.4 - 4.83i)T + (81.6 + 52.4i)T^{2} \) |
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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
−11.21307292250255534916069868992, −10.03759542287496109270585269572, −9.242916391988747236663618270500, −8.236848101904343687142234343150, −7.51250196568621728477111855476, −6.51795401220636830561896015089, −4.96551265695298970964244696475, −3.92430112745539309487336318795, −2.56648427731104082908902400740, −0.75429807301314545796581869902,
1.61181218093261342221016459989, 3.34933544588960172250235539939, 4.73730802267877799863602199843, 5.90491113669307164564141963667, 6.87060623217195997956984940688, 7.945826420161867728657865732545, 8.507884289218058713439837207570, 9.561925214425152028599569417564, 10.51233897437068450263801878491, 11.30394012807370980601200547549
