| L(s) = 1 | + (−200. − 137. i)3-s + 3.63e3i·5-s + 2.32e4·7-s + (2.11e4 + 5.51e4i)9-s + 6.24e4i·11-s + 1.70e5·13-s + (4.99e5 − 7.27e5i)15-s − 2.66e6i·17-s + 7.66e5·19-s + (−4.65e6 − 3.19e6i)21-s − 1.40e6i·23-s − 3.41e6·25-s + (3.34e6 − 1.39e7i)27-s − 4.83e6i·29-s + 4.18e7·31-s + ⋯ |
| L(s) = 1 | + (−0.824 − 0.566i)3-s + 1.16i·5-s + 1.38·7-s + (0.358 + 0.933i)9-s + 0.387i·11-s + 0.458·13-s + (0.657 − 0.957i)15-s − 1.87i·17-s + 0.309·19-s + (−1.13 − 0.782i)21-s − 0.218i·23-s − 0.349·25-s + (0.233 − 0.972i)27-s − 0.235i·29-s + 1.46·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (0.824 + 0.566i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(1.997243454\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.997243454\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (200. + 137. i)T \) |
| good | 5 | \( 1 - 3.63e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 2.32e4T + 2.82e8T^{2} \) |
| 11 | \( 1 - 6.24e4iT - 2.59e10T^{2} \) |
| 13 | \( 1 - 1.70e5T + 1.37e11T^{2} \) |
| 17 | \( 1 + 2.66e6iT - 2.01e12T^{2} \) |
| 19 | \( 1 - 7.66e5T + 6.13e12T^{2} \) |
| 23 | \( 1 + 1.40e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 + 4.83e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 4.18e7T + 8.19e14T^{2} \) |
| 37 | \( 1 + 5.01e7T + 4.80e15T^{2} \) |
| 41 | \( 1 + 1.49e8iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 1.98e8T + 2.16e16T^{2} \) |
| 47 | \( 1 - 1.55e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 4.21e7iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 2.92e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 5.30e8T + 7.13e17T^{2} \) |
| 67 | \( 1 - 5.22e8T + 1.82e18T^{2} \) |
| 71 | \( 1 + 5.71e8iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 2.18e9T + 4.29e18T^{2} \) |
| 79 | \( 1 + 1.96e9T + 9.46e18T^{2} \) |
| 83 | \( 1 + 2.18e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 2.38e8iT - 3.11e19T^{2} \) |
| 97 | \( 1 + 8.84e9T + 7.37e19T^{2} \) |
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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
−10.88964668190989640747117188308, −9.922541713103025994284676183028, −8.327405013977309521493744429408, −7.32665656466046826236131285085, −6.72305272954667597238314049383, −5.41734753088396967498813547916, −4.55787359187039846350009073756, −2.83007677399952487012573092861, −1.76115279320876141605324021915, −0.58123278671160711818418598925,
0.927543187069921424963790514847, 1.57833705181264508389992955616, 3.72754170224257722961055606495, 4.70139759589804764574248632708, 5.36939854324026970416066476663, 6.43245027789842345029052175845, 8.177632367459978978261183245544, 8.611231984976783490433659982828, 9.942767134437899599542994623760, 10.90531547772054789706196495803
