L(s) = 1 | + (2.10 − 1.65i)2-s + (0.814 − 0.580i)3-s + (1.21 − 5.02i)4-s + (1.23 + 0.238i)5-s + (0.754 − 2.56i)6-s + (0.245 + 2.63i)7-s + (−3.52 − 7.70i)8-s + (0.327 − 0.945i)9-s + (3.00 − 1.54i)10-s + (−1.73 + 2.20i)11-s + (−1.91 − 4.79i)12-s + (−0.367 + 0.572i)13-s + (4.87 + 5.13i)14-s + (1.14 − 0.524i)15-s + (−10.9 − 5.66i)16-s + (−4.01 + 3.82i)17-s + ⋯ |
L(s) = 1 | + (1.48 − 1.17i)2-s + (0.470 − 0.334i)3-s + (0.608 − 2.51i)4-s + (0.554 + 0.106i)5-s + (0.307 − 1.04i)6-s + (0.0926 + 0.995i)7-s + (−1.24 − 2.72i)8-s + (0.109 − 0.315i)9-s + (0.949 − 0.489i)10-s + (−0.523 + 0.665i)11-s + (−0.554 − 1.38i)12-s + (−0.101 + 0.158i)13-s + (1.30 + 1.37i)14-s + (0.296 − 0.135i)15-s + (−2.74 − 1.41i)16-s + (−0.974 + 0.928i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.288 + 0.957i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.17493 - 2.92626i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.17493 - 2.92626i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.814 + 0.580i)T \) |
| 7 | \( 1 + (-0.245 - 2.63i)T \) |
| 23 | \( 1 + (4.52 + 1.57i)T \) |
good | 2 | \( 1 + (-2.10 + 1.65i)T + (0.471 - 1.94i)T^{2} \) |
| 5 | \( 1 + (-1.23 - 0.238i)T + (4.64 + 1.85i)T^{2} \) |
| 11 | \( 1 + (1.73 - 2.20i)T + (-2.59 - 10.6i)T^{2} \) |
| 13 | \( 1 + (0.367 - 0.572i)T + (-5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (4.01 - 3.82i)T + (0.808 - 16.9i)T^{2} \) |
| 19 | \( 1 + (-3.02 - 2.88i)T + (0.904 + 18.9i)T^{2} \) |
| 29 | \( 1 + (-5.17 - 1.52i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-0.648 + 6.79i)T + (-30.4 - 5.86i)T^{2} \) |
| 37 | \( 1 + (8.22 + 2.84i)T + (29.0 + 22.8i)T^{2} \) |
| 41 | \( 1 + (-4.29 - 3.71i)T + (5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-2.19 - 1.00i)T + (28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 + (-6.70 + 3.87i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (8.55 - 0.407i)T + (52.7 - 5.03i)T^{2} \) |
| 59 | \( 1 + (1.80 + 3.50i)T + (-34.2 + 48.0i)T^{2} \) |
| 61 | \( 1 + (-3.96 + 5.57i)T + (-19.9 - 57.6i)T^{2} \) |
| 67 | \( 1 + (-2.60 + 6.51i)T + (-48.4 - 46.2i)T^{2} \) |
| 71 | \( 1 + (-0.464 - 3.22i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (-11.1 - 2.71i)T + (64.8 + 33.4i)T^{2} \) |
| 79 | \( 1 + (-16.7 - 0.796i)T + (78.6 + 7.50i)T^{2} \) |
| 83 | \( 1 + (7.34 + 8.48i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-3.21 + 0.307i)T + (87.3 - 16.8i)T^{2} \) |
| 97 | \( 1 + (-4.46 + 5.15i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.91420201057082637290335843535, −10.03851072466728902371079371355, −9.353243160209406833698305043077, −8.041893593393791738902929550407, −6.47866427266064229206849435326, −5.83002500919684687222291452665, −4.80402632952259391535415148709, −3.70978892922333476768959741511, −2.37482915768328012954855720251, −1.94331631439217721418943634024,
2.65743043576013608243784228838, 3.68886762136265758390486737450, 4.71313007914754714167897181275, 5.43372816343667295572293752536, 6.56960255526222505427692907660, 7.37619186969558578146323177997, 8.175738554109199890647702586777, 9.216625616175855207416191372418, 10.46294485178714046776443051537, 11.45365396892524957345490055624