| L(s) = 1 | + (−1.23 + 3.80i)2-s + (−12.9 − 9.40i)4-s + (−67.8 + 22.0i)5-s + (−151. + 209. i)7-s + (51.7 − 37.6i)8-s − 285. i·10-s + (−350. + 194. i)11-s + (−450. − 146. i)13-s + (−607. − 836. i)14-s + (79.1 + 243. i)16-s + (669. + 2.05e3i)17-s + (−539. − 742. i)19-s + (1.08e3 + 352. i)20-s + (−307. − 1.57e3i)22-s + 2.68e3i·23-s + ⋯ |
| L(s) = 1 | + (−0.218 + 0.672i)2-s + (−0.404 − 0.293i)4-s + (−1.21 + 0.394i)5-s + (−1.17 + 1.61i)7-s + (0.286 − 0.207i)8-s − 0.902i·10-s + (−0.874 + 0.485i)11-s + (−0.739 − 0.240i)13-s + (−0.828 − 1.14i)14-s + (0.0772 + 0.237i)16-s + (0.561 + 1.72i)17-s + (−0.342 − 0.472i)19-s + (0.606 + 0.197i)20-s + (−0.135 − 0.694i)22-s + 1.05i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.725 + 0.688i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.1619545601\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1619545601\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.23 - 3.80i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (350. - 194. i)T \) |
| good | 5 | \( 1 + (67.8 - 22.0i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (151. - 209. i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (450. + 146. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-669. - 2.05e3i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (539. + 742. i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 - 2.68e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (1.66e3 + 1.20e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-2.19e3 + 6.74e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (2.14e3 + 1.55e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (6.59e3 - 4.79e3i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 7.28e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-7.52e3 - 1.03e4i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (-2.90e4 - 9.45e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (-1.12e4 + 1.54e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (1.23e4 - 4.02e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 - 1.15e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + (-7.19e3 + 2.33e3i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (3.11e4 - 4.28e4i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-6.22e4 - 2.02e4i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (1.26e4 + 3.90e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 1.21e5iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (1.91e4 - 5.89e4i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−12.50954117013571045800200937232, −11.62889383229429315950440273498, −10.28263821789926598490883143715, −9.394964645559642378216712018083, −8.252938028975878440510233094231, −7.52134026675314985942531581459, −6.29072578781137069714866371918, −5.36313404635692896371933827650, −3.78385986771811975512037999270, −2.50808500721010216439633105790,
0.094105396702100308849627834559, 0.63358732960665126415564386549, 2.95415577819468774311060123074, 3.87656582504216019574159095314, 4.94036194687743028620678867780, 6.97602679237093430790042972289, 7.61707791606914591775155043080, 8.778990648025256262341907842076, 10.06248122814694461210325694679, 10.56427802710375998132730370651
