| L(s) = 1 | + (−0.700 + 2.09i)2-s + (0.632 − 0.774i)3-s + (−2.31 − 1.73i)4-s + (−0.365 − 1.48i)5-s + (1.18 + 1.86i)6-s + (−0.0438 − 0.543i)7-s + (1.62 − 1.11i)8-s + (−0.200 − 0.979i)9-s + (3.37 + 0.272i)10-s + (−2.12 − 0.434i)11-s + (−2.80 + 0.691i)12-s + (−3.52 − 0.751i)13-s + (1.17 + 0.288i)14-s + (−1.38 − 0.655i)15-s + (−0.397 − 1.37i)16-s + (−8.02 + 0.647i)17-s + ⋯ |
| L(s) = 1 | + (−0.495 + 1.48i)2-s + (0.365 − 0.447i)3-s + (−1.15 − 0.868i)4-s + (−0.163 − 0.664i)5-s + (0.482 + 0.763i)6-s + (−0.0165 − 0.205i)7-s + (0.572 − 0.395i)8-s + (−0.0666 − 0.326i)9-s + (1.06 + 0.0860i)10-s + (−0.641 − 0.130i)11-s + (−0.810 + 0.199i)12-s + (−0.978 − 0.208i)13-s + (0.312 + 0.0770i)14-s + (−0.356 − 0.169i)15-s + (−0.0992 − 0.342i)16-s + (−1.94 + 0.157i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 507 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.225 + 0.974i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.311329 - 0.247513i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.311329 - 0.247513i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.632 + 0.774i)T \) |
| 13 | \( 1 + (3.52 + 0.751i)T \) |
| good | 2 | \( 1 + (0.700 - 2.09i)T + (-1.59 - 1.20i)T^{2} \) |
| 5 | \( 1 + (0.365 + 1.48i)T + (-4.42 + 2.32i)T^{2} \) |
| 7 | \( 1 + (0.0438 + 0.543i)T + (-6.90 + 1.12i)T^{2} \) |
| 11 | \( 1 + (2.12 + 0.434i)T + (10.1 + 4.31i)T^{2} \) |
| 17 | \( 1 + (8.02 - 0.647i)T + (16.7 - 2.72i)T^{2} \) |
| 19 | \( 1 + (-0.698 - 0.403i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.423 + 0.733i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.60 + 2.20i)T + (23.1 + 17.4i)T^{2} \) |
| 31 | \( 1 + (-1.68 + 3.20i)T + (-17.6 - 25.5i)T^{2} \) |
| 37 | \( 1 + (6.63 - 0.267i)T + (36.8 - 2.97i)T^{2} \) |
| 41 | \( 1 + (-7.85 - 6.41i)T + (8.20 + 40.1i)T^{2} \) |
| 43 | \( 1 + (-0.260 + 6.45i)T + (-42.8 - 3.46i)T^{2} \) |
| 47 | \( 1 + (-1.35 + 0.164i)T + (45.6 - 11.2i)T^{2} \) |
| 53 | \( 1 + (-0.922 - 1.33i)T + (-18.7 + 49.5i)T^{2} \) |
| 59 | \( 1 + (8.50 + 2.46i)T + (49.8 + 31.5i)T^{2} \) |
| 61 | \( 1 + (0.298 + 0.629i)T + (-38.5 + 47.2i)T^{2} \) |
| 67 | \( 1 + (-3.90 - 5.19i)T + (-18.6 + 64.3i)T^{2} \) |
| 71 | \( 1 + (1.74 - 10.7i)T + (-67.3 - 22.4i)T^{2} \) |
| 73 | \( 1 + (-2.71 + 3.06i)T + (-8.79 - 72.4i)T^{2} \) |
| 79 | \( 1 + (1.46 + 12.0i)T + (-76.7 + 18.9i)T^{2} \) |
| 83 | \( 1 + (-3.95 - 1.49i)T + (62.1 + 55.0i)T^{2} \) |
| 89 | \( 1 + (-13.9 + 8.04i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (1.71 + 1.64i)T + (3.90 + 96.9i)T^{2} \) |
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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
−10.44796311335660482800631914913, −9.290752808961283943927038274717, −8.750984999834592370376362302712, −7.87985185150847764739967699597, −7.25983112452005452602034233397, −6.35396126781391561339589296267, −5.29902611138880215450387407360, −4.35175825302342410621302338773, −2.42884056100059021525270536356, −0.24585509024007477958229160137,
2.14167380519978597017589464717, 2.83532809526908401943536193891, 3.97320132435962283896474845276, 5.05327523279823918031266355739, 6.71937834112430290634046558773, 7.73536613273853076833783178244, 9.039660678938104729832643438573, 9.304724431810252801973751435023, 10.58171684620945300491578122979, 10.76366869612218556075417490809
