| L(s) = 1 | + (0.766 + 0.642i)2-s + (−0.766 − 0.642i)3-s + (0.173 + 0.984i)4-s + (0.162 − 0.918i)5-s + (−0.173 − 0.984i)6-s + (0.211 + 2.63i)7-s + (−0.500 + 0.866i)8-s + (0.173 + 0.984i)9-s + (0.714 − 0.599i)10-s + (−1.19 − 2.07i)11-s + (0.500 − 0.866i)12-s + (0.920 + 5.22i)13-s + (−1.53 + 2.15i)14-s + (−0.714 + 0.599i)15-s + (−0.939 + 0.342i)16-s + (−0.0144 + 0.0819i)17-s + ⋯ |
| L(s) = 1 | + (0.541 + 0.454i)2-s + (−0.442 − 0.371i)3-s + (0.0868 + 0.492i)4-s + (0.0724 − 0.410i)5-s + (−0.0708 − 0.402i)6-s + (0.0798 + 0.996i)7-s + (−0.176 + 0.306i)8-s + (0.0578 + 0.328i)9-s + (0.226 − 0.189i)10-s + (−0.361 − 0.625i)11-s + (0.144 − 0.249i)12-s + (0.255 + 1.44i)13-s + (−0.409 + 0.576i)14-s + (−0.184 + 0.154i)15-s + (−0.234 + 0.0855i)16-s + (−0.00350 + 0.0198i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0246 - 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 798 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0246 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.12084 + 1.14884i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.12084 + 1.14884i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (0.766 + 0.642i)T \) |
| 7 | \( 1 + (-0.211 - 2.63i)T \) |
| 19 | \( 1 + (2.29 - 3.70i)T \) |
| good | 5 | \( 1 + (-0.162 + 0.918i)T + (-4.69 - 1.71i)T^{2} \) |
| 11 | \( 1 + (1.19 + 2.07i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.920 - 5.22i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (0.0144 - 0.0819i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (-2.74 - 1.00i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-2.63 - 0.958i)T + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + 1.56T + 31T^{2} \) |
| 37 | \( 1 + (-3.66 - 6.35i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (0.0547 - 0.310i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (4.27 + 3.58i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-1.75 - 9.98i)T + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-0.445 - 2.52i)T + (-49.8 + 18.1i)T^{2} \) |
| 59 | \( 1 + (0.348 - 1.97i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (4.85 + 1.76i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-8.12 + 6.81i)T + (11.6 - 65.9i)T^{2} \) |
| 71 | \( 1 + (1.17 + 0.986i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (0.667 + 0.560i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (-7.89 + 2.87i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (3.71 + 6.43i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.04 + 5.91i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (5.40 - 1.96i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.74580825722843009460051494408, −9.372714268645893722283109077053, −8.679319476011888980104242741554, −7.921143139941698640489442193660, −6.72640708365574205211703555075, −6.10589832538366177400942930828, −5.24944783843893609124772981382, −4.41863342498969782945792265767, −3.01856613728091184554102707429, −1.66628876134092517634652662876,
0.73240744692185670490991459144, 2.55814621078958241920260396753, 3.61100933621034991139296688767, 4.62589168953567483050637010394, 5.36235920426228191716528549739, 6.52201219339891181298712764454, 7.24047344126632119078174420746, 8.342300807710847742256989221684, 9.589660597392685141430488083625, 10.48151959933714202934469929247
