| 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
−10.64695909569090811678596868566, −10.18216207555789290540044580692, −9.063029896376336570530032458876, −8.264323667986509942122726391599, −7.23739761364636389984222222169, −6.09956079611613032440397621098, −5.46985801257027741471061906650, −3.66275867889792141474103930197, −2.57801750174595468454221458837, −0.62458855760597834300105912170,
1.31662213397821747665312933220, 3.27935695644629758016313306676, 4.64484883826693473547320787217, 5.77435289538214005031491832554, 6.78914681098745833868530507814, 7.40359265527152376160044023286, 8.640145218251877593607519671521, 9.523368242695089035873571507118, 10.51013086368577181057068616796, 10.73622530365481197036767133257
