| L(s) = 1 | − 6.29i·3-s + 22.7·5-s − 51.5i·7-s + 41.3·9-s + 50.8i·11-s − 160.·13-s − 143. i·15-s + 416.·17-s − 515. i·19-s − 324.·21-s − 15.8i·23-s − 107.·25-s − 770. i·27-s − 979.·29-s − 1.90e3i·31-s + ⋯ |
| L(s) = 1 | − 0.699i·3-s + 0.909·5-s − 1.05i·7-s + 0.510·9-s + 0.419i·11-s − 0.949·13-s − 0.636i·15-s + 1.44·17-s − 1.42i·19-s − 0.735·21-s − 0.0299i·23-s − 0.172·25-s − 1.05i·27-s − 1.16·29-s − 1.98i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 128 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.43273 - 1.43273i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.43273 - 1.43273i\) |
| \(L(3)\) |
|
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 | \( 1 + 6.29iT - 81T^{2} \) |
| 5 | \( 1 - 22.7T + 625T^{2} \) |
| 7 | \( 1 + 51.5iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 50.8iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 160.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 416.T + 8.35e4T^{2} \) |
| 19 | \( 1 + 515. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + 15.8iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 979.T + 7.07e5T^{2} \) |
| 31 | \( 1 + 1.90e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 657.T + 1.87e6T^{2} \) |
| 41 | \( 1 - 2.83e3T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.53e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 940. iT - 4.87e6T^{2} \) |
| 53 | \( 1 - 280.T + 7.89e6T^{2} \) |
| 59 | \( 1 - 5.03e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.88e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.42e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 8.92e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 4.47e3T + 2.83e7T^{2} \) |
| 79 | \( 1 - 6.83e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 4.78e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.70e3T + 6.27e7T^{2} \) |
| 97 | \( 1 - 1.89e3T + 8.85e7T^{2} \) |
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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
−12.69420838640110451803255725833, −11.40502078759957672829725348988, −10.05288616752259809783961423542, −9.554417087592971451643914022894, −7.64083150291039022344153996380, −7.12806499164660761852150917861, −5.75203187971830556995093797452, −4.29012543299645680334869544648, −2.31975226373503970964119884052, −0.894614068115663459867021985419,
1.82166409912513387541847726951, 3.44568607053140771631002752346, 5.17389556023166717671503771732, 5.89464443366090608889927948570, 7.54753574949955762895849368249, 8.972795168255405982888437432637, 9.812426121422127736490891531075, 10.49164910662089664398923406045, 12.05220286108066056442379500471, 12.70081196304685309276031880758
