| L(s) = 1 | + (0.668 − 1.15i)2-s + (−0.320 + 2.98i)3-s + (1.10 + 1.91i)4-s + (4.41 − 2.33i)5-s + (3.24 + 2.36i)6-s + (−7.10 − 4.10i)7-s + 8.30·8-s + (−8.79 − 1.91i)9-s + (0.244 − 6.68i)10-s + (−5.67 − 3.27i)11-s + (−6.06 + 2.68i)12-s + (−1.29 + 0.749i)13-s + (−9.50 + 5.49i)14-s + (5.56 + 13.9i)15-s + (1.13 − 1.97i)16-s + 15.1·17-s + ⋯ |
| L(s) = 1 | + (0.334 − 0.579i)2-s + (−0.106 + 0.994i)3-s + (0.276 + 0.478i)4-s + (0.883 − 0.467i)5-s + (0.540 + 0.394i)6-s + (−1.01 − 0.586i)7-s + 1.03·8-s + (−0.977 − 0.212i)9-s + (0.0244 − 0.668i)10-s + (−0.515 − 0.297i)11-s + (−0.505 + 0.223i)12-s + (−0.0998 + 0.0576i)13-s + (−0.679 + 0.392i)14-s + (0.370 + 0.928i)15-s + (0.0710 − 0.123i)16-s + 0.889·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 45 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.995 - 0.0980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.35022 + 0.0663292i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.35022 + 0.0663292i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.320 - 2.98i)T \) |
| 5 | \( 1 + (-4.41 + 2.33i)T \) |
| good | 2 | \( 1 + (-0.668 + 1.15i)T + (-2 - 3.46i)T^{2} \) |
| 7 | \( 1 + (7.10 + 4.10i)T + (24.5 + 42.4i)T^{2} \) |
| 11 | \( 1 + (5.67 + 3.27i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (1.29 - 0.749i)T + (84.5 - 146. i)T^{2} \) |
| 17 | \( 1 - 15.1T + 289T^{2} \) |
| 19 | \( 1 + 25.9T + 361T^{2} \) |
| 23 | \( 1 + (11.6 + 20.1i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-6.96 - 4.02i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 + (-22.5 - 38.9i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 - 62.8iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (-9.97 + 5.75i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-36.9 - 21.3i)T + (924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-8.25 + 14.2i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + 66.0T + 2.80e3T^{2} \) |
| 59 | \( 1 + (-0.373 + 0.215i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-15.7 + 27.3i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (83.1 - 47.9i)T + (2.24e3 - 3.88e3i)T^{2} \) |
| 71 | \( 1 - 84.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 63.5iT - 5.32e3T^{2} \) |
| 79 | \( 1 + (-9.06 + 15.7i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-50.4 + 87.4i)T + (-3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + 86.3iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (59.7 + 34.5i)T + (4.70e3 + 8.14e3i)T^{2} \) |
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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
−15.97725831242608645657249393281, −14.25731778201918151919542591099, −13.17377282573798993649088154887, −12.22778319945829412963398452829, −10.59556015753417140654719445380, −10.02032034144418641537961358714, −8.452766205842448678441882654944, −6.33927323467611824228149438624, −4.57551088411597430169418342083, −3.00557186732055200284956037568,
2.32491768760797074064579614219, 5.69629086903218768537891853689, 6.32653310458610361001259408018, 7.59979246113986030873425338546, 9.570212570434718229995477318692, 10.79282436270445560020480082005, 12.46333931691205312137833912488, 13.39634089776840289928643080861, 14.37795540010183591004370900837, 15.42955400021147141409651807485
