| L(s) = 1 | + (0.841 − 0.540i)2-s + (0.415 − 0.909i)4-s + (−1.78 + 0.522i)5-s + (−2.54 − 2.93i)7-s + (−0.142 − 0.989i)8-s + (−1.21 + 1.40i)10-s + (−1.81 − 1.16i)11-s + (4.55 − 5.25i)13-s + (−3.72 − 1.09i)14-s + (−0.654 − 0.755i)16-s + (−2.36 − 5.17i)17-s + (−0.810 + 1.77i)19-s + (−0.264 + 1.83i)20-s − 2.15·22-s + (4.40 + 1.89i)23-s + ⋯ |
| L(s) = 1 | + (0.594 − 0.382i)2-s + (0.207 − 0.454i)4-s + (−0.796 + 0.233i)5-s + (−0.961 − 1.11i)7-s + (−0.0503 − 0.349i)8-s + (−0.384 + 0.443i)10-s + (−0.546 − 0.351i)11-s + (1.26 − 1.45i)13-s + (−0.996 − 0.292i)14-s + (−0.163 − 0.188i)16-s + (−0.573 − 1.25i)17-s + (−0.186 + 0.407i)19-s + (−0.0590 + 0.410i)20-s − 0.459·22-s + (0.918 + 0.394i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 414 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.555 + 0.831i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.595598 - 1.11426i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.595598 - 1.11426i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-4.40 - 1.89i)T \) |
| good | 5 | \( 1 + (1.78 - 0.522i)T + (4.20 - 2.70i)T^{2} \) |
| 7 | \( 1 + (2.54 + 2.93i)T + (-0.996 + 6.92i)T^{2} \) |
| 11 | \( 1 + (1.81 + 1.16i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (-4.55 + 5.25i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (2.36 + 5.17i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.810 - 1.77i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-0.0455 - 0.0997i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (-1.21 - 8.44i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-1.65 - 0.487i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (-7.89 + 2.31i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-0.147 + 1.02i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 + 3.27T + 47T^{2} \) |
| 53 | \( 1 + (7.20 + 8.31i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (1.62 - 1.88i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (-0.899 - 6.25i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (3.07 - 1.97i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (-0.478 + 0.307i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-5.79 + 12.6i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (-9.59 + 11.0i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (-9.91 - 2.91i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (-0.730 + 5.07i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (-6.38 + 1.87i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.79496906685054751406662431193, −10.47251948174859708004544581869, −9.229509727864593970406511906120, −7.954908088533050074337160680942, −7.12467728691886501049345434338, −6.12074426599579208023248918468, −4.90876252025696442658287376871, −3.56986006795664743328949383669, −3.13412567513716610528838814406, −0.67365840120219943113683167781,
2.36127175084848787855739697153, 3.74926837812827055996931266916, 4.58847026335952082043984023068, 6.06185800347054127264486960223, 6.51060784408171566702240440688, 7.85401706355116119944569969058, 8.733369280129842838192933421918, 9.444331285288005198368028309214, 10.98400609799333857495592826822, 11.59047116692378112912550001772
