| L(s) = 1 | + (0.913 + 0.406i)2-s + (0.978 + 0.207i)3-s + (0.669 + 0.743i)4-s + (0.0926 − 0.881i)5-s + (0.809 + 0.587i)6-s + (1.77 − 1.95i)7-s + (0.309 + 0.951i)8-s + (0.913 + 0.406i)9-s + (0.443 − 0.768i)10-s + (3.02 − 1.35i)11-s + (0.499 + 0.866i)12-s + (−3.78 + 2.75i)13-s + (2.42 − 1.06i)14-s + (0.274 − 0.843i)15-s + (−0.104 + 0.994i)16-s + (−3.02 + 1.34i)17-s + ⋯ |
| L(s) = 1 | + (0.645 + 0.287i)2-s + (0.564 + 0.120i)3-s + (0.334 + 0.371i)4-s + (0.0414 − 0.394i)5-s + (0.330 + 0.239i)6-s + (0.672 − 0.740i)7-s + (0.109 + 0.336i)8-s + (0.304 + 0.135i)9-s + (0.140 − 0.242i)10-s + (0.913 − 0.407i)11-s + (0.144 + 0.249i)12-s + (−1.05 + 0.763i)13-s + (0.647 − 0.284i)14-s + (0.0707 − 0.217i)15-s + (−0.0261 + 0.248i)16-s + (−0.733 + 0.326i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.975 - 0.221i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.54459 + 0.285224i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.54459 + 0.285224i\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + (-1.77 + 1.95i)T \) |
| 11 | \( 1 + (-3.02 + 1.35i)T \) |
| good | 5 | \( 1 + (-0.0926 + 0.881i)T + (-4.89 - 1.03i)T^{2} \) |
| 13 | \( 1 + (3.78 - 2.75i)T + (4.01 - 12.3i)T^{2} \) |
| 17 | \( 1 + (3.02 - 1.34i)T + (11.3 - 12.6i)T^{2} \) |
| 19 | \( 1 + (-2.58 + 2.87i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (2.93 + 5.07i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.05 - 6.33i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-1.02 - 9.77i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (0.203 - 0.0432i)T + (33.8 - 15.0i)T^{2} \) |
| 41 | \( 1 + (1.71 + 5.27i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + 6.38T + 43T^{2} \) |
| 47 | \( 1 + (7.44 - 8.27i)T + (-4.91 - 46.7i)T^{2} \) |
| 53 | \( 1 + (1.23 + 11.7i)T + (-51.8 + 11.0i)T^{2} \) |
| 59 | \( 1 + (-1.40 - 1.56i)T + (-6.16 + 58.6i)T^{2} \) |
| 61 | \( 1 + (0.992 - 9.44i)T + (-59.6 - 12.6i)T^{2} \) |
| 67 | \( 1 + (-0.784 + 1.35i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (3.69 + 2.68i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (7.69 + 8.54i)T + (-7.63 + 72.6i)T^{2} \) |
| 79 | \( 1 + (-13.9 - 6.22i)T + (52.8 + 58.7i)T^{2} \) |
| 83 | \( 1 + (13.5 + 9.87i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 + (5.12 + 8.87i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (2.58 - 1.88i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.19958574282932377564836252740, −10.24790449158677179394679083917, −9.014990794282362871044790875814, −8.460957978902099202418050916909, −7.18770278771196479103446097909, −6.66474185286287026384980741546, −4.98511690879679572439992185714, −4.45844302918023240397116576674, −3.26137360896321990433636070208, −1.68990221142828122072244122108,
1.85876407173987198932976378450, 2.84979242217123840027529063036, 4.11857860362473674765994827989, 5.19089286610948456224560100950, 6.25245522131323910754808013314, 7.38026654691279314806082999514, 8.176478806475292062341791717491, 9.476681834586393052145588908253, 9.974961059211867688459002505462, 11.38805510062846781715261947851
