| L(s) = 1 | + (0.988 − 0.149i)2-s + (1.27 + 0.871i)3-s + (0.955 − 0.294i)4-s + (0.270 − 3.60i)5-s + (1.39 + 0.670i)6-s + (0.900 − 0.433i)8-s + (−0.222 − 0.566i)9-s + (−0.270 − 3.60i)10-s + (−1.32 + 3.36i)11-s + (1.47 + 0.455i)12-s + (0.304 + 0.381i)13-s + (3.48 − 4.37i)15-s + (0.826 − 0.563i)16-s + (4.00 − 3.71i)17-s + (−0.304 − 0.527i)18-s + (1.14 − 1.99i)19-s + ⋯ |
| L(s) = 1 | + (0.699 − 0.105i)2-s + (0.737 + 0.502i)3-s + (0.477 − 0.147i)4-s + (0.120 − 1.61i)5-s + (0.568 + 0.273i)6-s + (0.318 − 0.153i)8-s + (−0.0741 − 0.188i)9-s + (−0.0854 − 1.14i)10-s + (−0.398 + 1.01i)11-s + (0.426 + 0.131i)12-s + (0.0844 + 0.105i)13-s + (0.899 − 1.12i)15-s + (0.206 − 0.140i)16-s + (0.970 − 0.900i)17-s + (−0.0717 − 0.124i)18-s + (0.263 − 0.456i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 686 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.767 + 0.641i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.72715 - 0.990303i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.72715 - 0.990303i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.988 + 0.149i)T \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + (-1.27 - 0.871i)T + (1.09 + 2.79i)T^{2} \) |
| 5 | \( 1 + (-0.270 + 3.60i)T + (-4.94 - 0.745i)T^{2} \) |
| 11 | \( 1 + (1.32 - 3.36i)T + (-8.06 - 7.48i)T^{2} \) |
| 13 | \( 1 + (-0.304 - 0.381i)T + (-2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-4.00 + 3.71i)T + (1.27 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.14 + 1.99i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-3.19 - 2.96i)T + (1.71 + 22.9i)T^{2} \) |
| 29 | \( 1 + (-1.36 - 5.96i)T + (-26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-0.268 - 0.464i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (9.33 + 2.87i)T + (30.5 + 20.8i)T^{2} \) |
| 41 | \( 1 + (5.50 - 2.65i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-8.23 - 3.96i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (11.1 - 1.68i)T + (44.9 - 13.8i)T^{2} \) |
| 53 | \( 1 + (3.66 - 1.13i)T + (43.7 - 29.8i)T^{2} \) |
| 59 | \( 1 + (-0.275 - 3.67i)T + (-58.3 + 8.79i)T^{2} \) |
| 61 | \( 1 + (-9.66 - 2.98i)T + (50.4 + 34.3i)T^{2} \) |
| 67 | \( 1 + (-5.78 - 10.0i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.229 - 1.00i)T + (-63.9 - 30.8i)T^{2} \) |
| 73 | \( 1 + (-0.823 - 0.124i)T + (69.7 + 21.5i)T^{2} \) |
| 79 | \( 1 + (-0.637 + 1.10i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.98 - 2.49i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-4.54 - 11.5i)T + (-65.2 + 60.5i)T^{2} \) |
| 97 | \( 1 + 7.16T + 97T^{2} \) |
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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
−10.12266213257198711311428556016, −9.460884630442902921571807167755, −8.843037736797605276891933405245, −7.87839339998093698737449442982, −6.86189240195504547736657181199, −5.30126159665599580851548238825, −4.97762072043294711848407606839, −3.90595053259449536587033455239, −2.83190003070011403899041142407, −1.31363378886868994353365714276,
2.02033815602232784691184295936, 3.07357863443018009588152140402, 3.54597293281179284895384968657, 5.30859493170870162328736247052, 6.21302595155861089030042673819, 6.98385911829644420200603389156, 7.899700381023041429613087756099, 8.438130460495433927348098027627, 9.999891357960533446745754200598, 10.67964445441946218583660368605
