| L(s) = 1 | + (−1.83 + 0.175i)2-s + (−0.690 − 0.723i)3-s + (1.38 − 0.267i)4-s + (−1.78 − 0.917i)5-s + (1.39 + 1.20i)6-s + (−2.33 + 1.23i)7-s + (1.04 − 0.306i)8-s + (−0.0475 + 0.998i)9-s + (3.43 + 1.37i)10-s + (0.533 − 5.58i)11-s + (−1.14 − 0.818i)12-s + (−3.84 + 0.553i)13-s + (4.08 − 2.68i)14-s + (0.564 + 1.92i)15-s + (−4.48 + 1.79i)16-s + (−0.0974 − 0.281i)17-s + ⋯ |
| L(s) = 1 | + (−1.30 + 0.124i)2-s + (−0.398 − 0.417i)3-s + (0.692 − 0.133i)4-s + (−0.796 − 0.410i)5-s + (0.569 + 0.493i)6-s + (−0.883 + 0.467i)7-s + (0.368 − 0.108i)8-s + (−0.0158 + 0.332i)9-s + (1.08 + 0.434i)10-s + (0.160 − 1.68i)11-s + (−0.331 − 0.236i)12-s + (−1.06 + 0.153i)13-s + (1.09 − 0.718i)14-s + (0.145 + 0.496i)15-s + (−1.12 + 0.448i)16-s + (−0.0236 − 0.0683i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 483 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.470 - 0.882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.210134 + 0.126078i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.210134 + 0.126078i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.690 + 0.723i)T \) |
| 7 | \( 1 + (2.33 - 1.23i)T \) |
| 23 | \( 1 + (-2.41 - 4.14i)T \) |
| good | 2 | \( 1 + (1.83 - 0.175i)T + (1.96 - 0.378i)T^{2} \) |
| 5 | \( 1 + (1.78 + 0.917i)T + (2.90 + 4.07i)T^{2} \) |
| 11 | \( 1 + (-0.533 + 5.58i)T + (-10.8 - 2.08i)T^{2} \) |
| 13 | \( 1 + (3.84 - 0.553i)T + (12.4 - 3.66i)T^{2} \) |
| 17 | \( 1 + (0.0974 + 0.281i)T + (-13.3 + 10.5i)T^{2} \) |
| 19 | \( 1 + (0.874 - 2.52i)T + (-14.9 - 11.7i)T^{2} \) |
| 29 | \( 1 + (-0.420 + 0.484i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-4.06 - 0.986i)T + (27.5 + 14.2i)T^{2} \) |
| 37 | \( 1 + (-11.4 - 0.544i)T + (36.8 + 3.51i)T^{2} \) |
| 41 | \( 1 + (-2.97 - 4.62i)T + (-17.0 + 37.2i)T^{2} \) |
| 43 | \( 1 + (3.58 - 12.2i)T + (-36.1 - 23.2i)T^{2} \) |
| 47 | \( 1 + (5.76 - 3.32i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3.24 - 4.13i)T + (-12.4 - 51.5i)T^{2} \) |
| 59 | \( 1 + (0.264 - 0.660i)T + (-42.7 - 40.7i)T^{2} \) |
| 61 | \( 1 + (-9.28 - 8.85i)T + (2.90 + 60.9i)T^{2} \) |
| 67 | \( 1 + (-8.35 + 5.94i)T + (21.9 - 63.3i)T^{2} \) |
| 71 | \( 1 + (-0.457 + 1.00i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (-2.56 - 13.3i)T + (-67.7 + 27.1i)T^{2} \) |
| 79 | \( 1 + (6.90 + 8.77i)T + (-18.6 + 76.7i)T^{2} \) |
| 83 | \( 1 + (8.84 + 5.68i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-0.297 - 1.22i)T + (-79.1 + 40.7i)T^{2} \) |
| 97 | \( 1 + (2.10 - 1.35i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.23064978734318337649363459494, −9.958900531010470497615594101694, −9.345622711670525206745281275353, −8.296798565617513197778152210290, −7.87362265203263963742230049512, −6.72245344495909528529222111399, −5.85155448021326849097343377736, −4.41463315079376372120140019944, −2.90991552955182256877434804822, −0.941931738793622525810298550011,
0.32277344615339673292904880482, 2.44250121995398759366479715546, 4.04895300855962466319063105594, 4.91259976819369502355396161277, 6.82309339471108583804171692040, 7.14692191926910940481536774107, 8.154870513495749132368227051631, 9.426789032206040706361151374645, 9.839150531127549784974284899183, 10.54156598887482441449049995011
