| L(s) = 1 | + (−0.576 − 0.0504i)2-s + (−1.73 − 0.0483i)3-s + (−1.63 − 0.289i)4-s + (2.18 + 0.474i)5-s + (0.996 + 0.115i)6-s + (2.78 + 1.94i)7-s + (2.05 + 0.549i)8-s + (2.99 + 0.167i)9-s + (−1.23 − 0.384i)10-s + (1.55 − 4.27i)11-s + (2.82 + 0.579i)12-s + (0.357 + 4.08i)13-s + (−1.50 − 1.26i)14-s + (−3.76 − 0.927i)15-s + (1.97 + 0.718i)16-s + (0.284 − 0.0763i)17-s + ⋯ |
| L(s) = 1 | + (−0.407 − 0.0356i)2-s + (−0.999 − 0.0278i)3-s + (−0.819 − 0.144i)4-s + (0.977 + 0.212i)5-s + (0.406 + 0.0470i)6-s + (1.05 + 0.735i)7-s + (0.724 + 0.194i)8-s + (0.998 + 0.0557i)9-s + (−0.391 − 0.121i)10-s + (0.469 − 1.29i)11-s + (0.815 + 0.167i)12-s + (0.0990 + 1.13i)13-s + (−0.402 − 0.337i)14-s + (−0.970 − 0.239i)15-s + (0.493 + 0.179i)16-s + (0.0690 − 0.0185i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.967 - 0.251i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.734716 + 0.0938917i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.734716 + 0.0938917i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.73 + 0.0483i)T \) |
| 5 | \( 1 + (-2.18 - 0.474i)T \) |
| good | 2 | \( 1 + (0.576 + 0.0504i)T + (1.96 + 0.347i)T^{2} \) |
| 7 | \( 1 + (-2.78 - 1.94i)T + (2.39 + 6.57i)T^{2} \) |
| 11 | \( 1 + (-1.55 + 4.27i)T + (-8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.357 - 4.08i)T + (-12.8 + 2.25i)T^{2} \) |
| 17 | \( 1 + (-0.284 + 0.0763i)T + (14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (4.26 - 2.46i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.869 - 1.24i)T + (-7.86 + 21.6i)T^{2} \) |
| 29 | \( 1 + (-5.57 + 4.67i)T + (5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-0.591 + 3.35i)T + (-29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-1.28 - 4.81i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 + (2.42 - 2.89i)T + (-7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (4.02 + 8.63i)T + (-27.6 + 32.9i)T^{2} \) |
| 47 | \( 1 + (-1.09 + 1.56i)T + (-16.0 - 44.1i)T^{2} \) |
| 53 | \( 1 + (1.39 - 1.39i)T - 53iT^{2} \) |
| 59 | \( 1 + (9.34 - 3.40i)T + (45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (0.249 + 1.41i)T + (-57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (9.46 - 0.827i)T + (65.9 - 11.6i)T^{2} \) |
| 71 | \( 1 + (-0.728 - 0.420i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-0.166 + 0.619i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-9.56 - 11.3i)T + (-13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-0.830 + 9.49i)T + (-81.7 - 14.4i)T^{2} \) |
| 89 | \( 1 + (6.19 + 10.7i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (4.11 - 1.91i)T + (62.3 - 74.3i)T^{2} \) |
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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
−13.44337390741762781310791637679, −12.03224547506982448632805755812, −11.16999541269960228817554125508, −10.22545344173753233448226557282, −9.140309155909118665399312845050, −8.261318151489478857314326218754, −6.42541140247425808333390930021, −5.56086963787555913609080910510, −4.42798481910687659664928397765, −1.59977034177561869579399049516,
1.31921066887543157482505360045, 4.48952747111282419838405414491, 5.09761598722200327164549661474, 6.67977728891823145432704002997, 7.88139468799491406267663638680, 9.175766394844387901448350210885, 10.27708354961034527784700250282, 10.74220175867077139756757169198, 12.38555451564025329329669321995, 13.00813183029887450136060802147
