| L(s) = 1 | − 46.7i·3-s + 230.·5-s − 3.73e3i·7-s − 2.18e3·9-s − 4.19e3i·11-s + 3.89e4·13-s − 1.07e4i·15-s − 1.48e5·17-s − 3.40e4i·19-s − 1.74e5·21-s + 3.88e5i·23-s − 3.37e5·25-s + 1.02e5i·27-s + 2.86e5·29-s + 6.05e5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s + 0.368·5-s − 1.55i·7-s − 0.333·9-s − 0.286i·11-s + 1.36·13-s − 0.212i·15-s − 1.77·17-s − 0.261i·19-s − 0.897·21-s + 1.38i·23-s − 0.864·25-s + 0.192i·27-s + 0.405·29-s + 0.655i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.2805682943\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.2805682943\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
| good | 5 | \( 1 - 230.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 3.73e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 4.19e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.89e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.48e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 3.40e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 3.88e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 2.86e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 6.05e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 3.48e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 2.59e5T + 7.98e12T^{2} \) |
| 43 | \( 1 + 4.88e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 5.00e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 9.11e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 8.08e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 1.17e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 3.08e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.64e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.37e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 3.38e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 4.02e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 1.10e8T + 3.93e15T^{2} \) |
| 97 | \( 1 - 1.13e8T + 7.83e15T^{2} \) |
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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
−10.46995838636596661489824322931, −9.175096721865118806441453443468, −8.362616842672525135395837526233, −7.23235795459038122589470294547, −6.65265459790834224744853100899, −5.60491512282874810715317024251, −4.24053112043315832707506153250, −3.40189264511526274069265482857, −1.87296968376196712957341708923, −1.03994807331796452003434563297,
0.05400726475186019482392799050, 1.82218131710247429814334470039, 2.60678178257344234361346091442, 3.91016395888860239245839229423, 4.97259447013839706765882715112, 5.96217924720292006172067226718, 6.60124837103241684782790457119, 8.401844453721192759857607984238, 8.752072872305110945841769913374, 9.646668548736218038682776035080
