| L(s) = 1 | + 452. i·3-s + 4.03e3·5-s − 2.65e4i·7-s − 1.45e5·9-s + 2.64e5i·11-s − 3.95e4·13-s + 1.82e6i·15-s + 3.44e5·17-s + 2.00e6i·19-s + 1.20e7·21-s + 2.18e6i·23-s + 6.54e6·25-s − 3.91e7i·27-s + 1.83e7·29-s + 2.39e7i·31-s + ⋯ |
| L(s) = 1 | + 1.86i·3-s + 1.29·5-s − 1.58i·7-s − 2.46·9-s + 1.64i·11-s − 0.106·13-s + 2.40i·15-s + 0.242·17-s + 0.809i·19-s + 2.94·21-s + 0.340i·23-s + 0.670·25-s − 2.72i·27-s + 0.893·29-s + 0.838i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(2.232441035\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.232441035\) |
| \(L(6)\) |
|
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 | \( 1 - 452. iT - 5.90e4T^{2} \) |
| 5 | \( 1 - 4.03e3T + 9.76e6T^{2} \) |
| 7 | \( 1 + 2.65e4iT - 2.82e8T^{2} \) |
| 11 | \( 1 - 2.64e5iT - 2.59e10T^{2} \) |
| 13 | \( 1 + 3.95e4T + 1.37e11T^{2} \) |
| 17 | \( 1 - 3.44e5T + 2.01e12T^{2} \) |
| 19 | \( 1 - 2.00e6iT - 6.13e12T^{2} \) |
| 23 | \( 1 - 2.18e6iT - 4.14e13T^{2} \) |
| 29 | \( 1 - 1.83e7T + 4.20e14T^{2} \) |
| 31 | \( 1 - 2.39e7iT - 8.19e14T^{2} \) |
| 37 | \( 1 - 1.22e8T + 4.80e15T^{2} \) |
| 41 | \( 1 - 8.19e7T + 1.34e16T^{2} \) |
| 43 | \( 1 - 1.09e8iT - 2.16e16T^{2} \) |
| 47 | \( 1 + 1.39e8iT - 5.25e16T^{2} \) |
| 53 | \( 1 + 2.40e8T + 1.74e17T^{2} \) |
| 59 | \( 1 - 1.70e8iT - 5.11e17T^{2} \) |
| 61 | \( 1 - 5.72e8T + 7.13e17T^{2} \) |
| 67 | \( 1 + 5.78e7iT - 1.82e18T^{2} \) |
| 71 | \( 1 - 2.82e7iT - 3.25e18T^{2} \) |
| 73 | \( 1 - 8.76e8T + 4.29e18T^{2} \) |
| 79 | \( 1 - 4.19e9iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 1.91e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 + 8.53e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + 3.19e9T + 7.37e19T^{2} \) |
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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
−10.27394002477993023002987313004, −9.915856837978562657011377371239, −9.421484065125028192568871049811, −7.951729888517986822973155698641, −6.66336045778791420313623609233, −5.46937076588712090893495533459, −4.56404655259993590005203792316, −3.93216155432261015336082391351, −2.63529362354785437209797721068, −1.25857408493894241801316029263,
0.41065378216111934703175553451, 1.34225435147690359156588408659, 2.48691435529334205555828484056, 2.73991079441716006562115107154, 5.40409373906740225703266890582, 6.00796588233583901332944987529, 6.47400001297448962764380725055, 7.922093237997804996061047992417, 8.683511496539573640508798167279, 9.407488495839454737267922691058
