| L(s) = 1 | + (−0.766 − 0.642i)2-s + (0.391 − 1.68i)3-s + (0.173 + 0.984i)4-s + (2.08 + 0.758i)5-s + (−1.38 + 1.04i)6-s + (0.173 − 0.984i)7-s + (0.500 − 0.866i)8-s + (−2.69 − 1.32i)9-s + (−1.10 − 1.92i)10-s + (2.50 − 0.912i)11-s + (1.72 + 0.0926i)12-s + (0.389 − 0.327i)13-s + (−0.766 + 0.642i)14-s + (2.09 − 3.22i)15-s + (−0.939 + 0.342i)16-s + (1.36 + 2.35i)17-s + ⋯ |
| L(s) = 1 | + (−0.541 − 0.454i)2-s + (0.226 − 0.974i)3-s + (0.0868 + 0.492i)4-s + (0.932 + 0.339i)5-s + (−0.565 + 0.424i)6-s + (0.0656 − 0.372i)7-s + (0.176 − 0.306i)8-s + (−0.897 − 0.440i)9-s + (−0.350 − 0.607i)10-s + (0.755 − 0.274i)11-s + (0.499 + 0.0267i)12-s + (0.108 − 0.0906i)13-s + (−0.204 + 0.171i)14-s + (0.541 − 0.831i)15-s + (−0.234 + 0.0855i)16-s + (0.329 + 0.571i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0442 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0442 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.903942 - 0.944842i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.903942 - 0.944842i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.766 + 0.642i)T \) |
| 3 | \( 1 + (-0.391 + 1.68i)T \) |
| 7 | \( 1 + (-0.173 + 0.984i)T \) |
| good | 5 | \( 1 + (-2.08 - 0.758i)T + (3.83 + 3.21i)T^{2} \) |
| 11 | \( 1 + (-2.50 + 0.912i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.389 + 0.327i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.36 - 2.35i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.46 + 5.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.481 - 2.72i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (0.825 + 0.692i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (0.526 + 2.98i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (4.07 + 7.06i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (7.78 - 6.52i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.03 - 1.46i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (0.0998 - 0.566i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 - 6.87T + 53T^{2} \) |
| 59 | \( 1 + (-11.6 - 4.22i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (2.03 - 11.5i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.89 + 2.42i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (-6.07 - 10.5i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.24 + 3.88i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.77 - 7.36i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (12.1 + 10.2i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (4.65 - 8.06i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (11.7 - 4.26i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.25391923457000419496620287155, −10.12823939062913709955651883672, −9.299347158747900578269291446687, −8.453406944713104532951382166811, −7.33947851209905699445118380750, −6.60326796563780041611992844164, −5.54018617615071547071098900229, −3.61895387625792401556928283848, −2.39212837847420304244162553554, −1.16877729696921529093321460350,
1.81251794859908399784859214719, 3.50439569425959482254491785942, 5.02037332655566707041824102522, 5.66836893675746545107316707802, 6.82104013012958925171365203152, 8.224774846614457902577521748564, 8.941347935891581038848871121619, 9.794986221393274732688702024516, 10.16300598546168254297218533831, 11.44264361168638809383498921411
