| L(s) = 1 | − 214.·3-s + 3.26e3i·5-s + 5.80e3i·7-s − 1.30e4·9-s − 2.71e5·11-s − 5.57e5i·13-s − 6.99e5i·15-s − 7.95e5·17-s + 4.01e6·19-s − 1.24e6i·21-s − 7.93e6i·23-s − 8.92e5·25-s + 1.54e7·27-s + 7.81e6i·29-s + 1.15e7i·31-s + ⋯ |
| L(s) = 1 | − 0.882·3-s + 1.04i·5-s + 0.345i·7-s − 0.221·9-s − 1.68·11-s − 1.50i·13-s − 0.921i·15-s − 0.560·17-s + 1.62·19-s − 0.304i·21-s − 1.23i·23-s − 0.0913·25-s + 1.07·27-s + 0.380i·29-s + 0.403i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.707 - 0.707i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.5558860947\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5558860947\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + 214.T + 5.90e4T^{2} \) |
| 5 | \( 1 - 3.26e3iT - 9.76e6T^{2} \) |
| 7 | \( 1 - 5.80e3iT - 2.82e8T^{2} \) |
| 11 | \( 1 + 2.71e5T + 2.59e10T^{2} \) |
| 13 | \( 1 + 5.57e5iT - 1.37e11T^{2} \) |
| 17 | \( 1 + 7.95e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 4.01e6T + 6.13e12T^{2} \) |
| 23 | \( 1 + 7.93e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 7.81e6iT - 4.20e14T^{2} \) |
| 31 | \( 1 - 1.15e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 1.07e8iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 5.82e7T + 1.34e16T^{2} \) |
| 43 | \( 1 + 7.58e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + 1.19e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 - 4.68e8iT - 1.74e17T^{2} \) |
| 59 | \( 1 - 8.13e8T + 5.11e17T^{2} \) |
| 61 | \( 1 + 4.39e8iT - 7.13e17T^{2} \) |
| 67 | \( 1 - 1.14e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 9.19e6iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 8.89e8T + 4.29e18T^{2} \) |
| 79 | \( 1 + 5.96e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 + 2.00e8T + 1.55e19T^{2} \) |
| 89 | \( 1 + 7.64e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 6.43e9T + 7.37e19T^{2} \) |
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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
−10.56289975430782832506624507955, −10.16109040372158087892050500388, −8.523430248191291708906736457423, −7.61461264305806792627538238797, −6.58919539774743624219503143329, −5.55270618378689062583974569285, −5.01613115594627024051108328158, −3.04958205886335981880485522087, −2.66372519884926624051966017998, −0.74378965659792302965774525757,
0.18381499270780500152385522133, 1.11530766382270203413704503954, 2.43987244670096111835724034919, 4.01472305750945746070312972475, 5.15457376812855627207653688124, 5.53551258475058756212709038174, 6.94336799145090970504210624384, 7.909219158140485819880922515561, 9.015667060620956152061164001971, 9.864911384254445516966399611209
