| L(s) = 1 | + (−0.913 + 0.406i)2-s + (−0.978 + 0.207i)3-s + (0.669 − 0.743i)4-s + (−0.0207 − 0.197i)5-s + (0.809 − 0.587i)6-s + (−2.07 − 1.63i)7-s + (−0.309 + 0.951i)8-s + (0.913 − 0.406i)9-s + (0.0993 + 0.172i)10-s + (−1.70 + 2.84i)11-s + (−0.499 + 0.866i)12-s + (2.44 + 1.77i)13-s + (2.56 + 0.650i)14-s + (0.0614 + 0.189i)15-s + (−0.104 − 0.994i)16-s + (4.83 + 2.15i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 + 0.287i)2-s + (−0.564 + 0.120i)3-s + (0.334 − 0.371i)4-s + (−0.00929 − 0.0884i)5-s + (0.330 − 0.239i)6-s + (−0.785 − 0.618i)7-s + (−0.109 + 0.336i)8-s + (0.304 − 0.135i)9-s + (0.0314 + 0.0544i)10-s + (−0.513 + 0.858i)11-s + (−0.144 + 0.249i)12-s + (0.676 + 0.491i)13-s + (0.685 + 0.173i)14-s + (0.0158 + 0.0488i)15-s + (−0.0261 − 0.248i)16-s + (1.17 + 0.522i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.175 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.521331 + 0.436401i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.521331 + 0.436401i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.913 - 0.406i)T \) |
| 3 | \( 1 + (0.978 - 0.207i)T \) |
| 7 | \( 1 + (2.07 + 1.63i)T \) |
| 11 | \( 1 + (1.70 - 2.84i)T \) |
| good | 5 | \( 1 + (0.0207 + 0.197i)T + (-4.89 + 1.03i)T^{2} \) |
| 13 | \( 1 + (-2.44 - 1.77i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-4.83 - 2.15i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (0.813 + 0.903i)T + (-1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (0.917 - 1.58i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.77 - 8.54i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (0.533 - 5.07i)T + (-30.3 - 6.44i)T^{2} \) |
| 37 | \( 1 + (11.3 + 2.40i)T + (33.8 + 15.0i)T^{2} \) |
| 41 | \( 1 + (2.45 - 7.54i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 10.2T + 43T^{2} \) |
| 47 | \( 1 + (-8.35 - 9.27i)T + (-4.91 + 46.7i)T^{2} \) |
| 53 | \( 1 + (-0.712 + 6.77i)T + (-51.8 - 11.0i)T^{2} \) |
| 59 | \( 1 + (1.36 - 1.51i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (-0.937 - 8.92i)T + (-59.6 + 12.6i)T^{2} \) |
| 67 | \( 1 + (7.93 + 13.7i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (3.72 - 2.70i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-3.80 + 4.22i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (-7.43 + 3.31i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (11.6 - 8.42i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + (-3.05 + 5.28i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.40 - 1.74i)T + (29.9 + 92.2i)T^{2} \) |
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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
−10.73622530365481197036767133257, −10.51013086368577181057068616796, −9.523368242695089035873571507118, −8.640145218251877593607519671521, −7.40359265527152376160044023286, −6.78914681098745833868530507814, −5.77435289538214005031491832554, −4.64484883826693473547320787217, −3.27935695644629758016313306676, −1.31662213397821747665312933220,
0.62458855760597834300105912170, 2.57801750174595468454221458837, 3.66275867889792141474103930197, 5.46985801257027741471061906650, 6.09956079611613032440397621098, 7.23739761364636389984222222169, 8.264323667986509942122726391599, 9.063029896376336570530032458876, 10.18216207555789290540044580692, 10.64695909569090811678596868566
