| L(s) = 1 | − 3.85i·3-s − 0.490·5-s + 2.64i·7-s − 5.87·9-s − 15.5i·11-s − 3.50·13-s + 1.89i·15-s − 24.1·17-s − 3.56i·19-s + 10.2·21-s − 19.5i·23-s − 24.7·25-s − 12.0i·27-s + 10.9·29-s + 21.1i·31-s + ⋯ |
| L(s) = 1 | − 1.28i·3-s − 0.0980·5-s + 0.377i·7-s − 0.652·9-s − 1.41i·11-s − 0.269·13-s + 0.126i·15-s − 1.42·17-s − 0.187i·19-s + 0.485·21-s − 0.851i·23-s − 0.990·25-s − 0.446i·27-s + 0.378·29-s + 0.683i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.9244832197\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9244832197\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 3 | \( 1 + 3.85iT - 9T^{2} \) |
| 5 | \( 1 + 0.490T + 25T^{2} \) |
| 11 | \( 1 + 15.5iT - 121T^{2} \) |
| 13 | \( 1 + 3.50T + 169T^{2} \) |
| 17 | \( 1 + 24.1T + 289T^{2} \) |
| 19 | \( 1 + 3.56iT - 361T^{2} \) |
| 23 | \( 1 + 19.5iT - 529T^{2} \) |
| 29 | \( 1 - 10.9T + 841T^{2} \) |
| 31 | \( 1 - 21.1iT - 961T^{2} \) |
| 37 | \( 1 + 58.4T + 1.36e3T^{2} \) |
| 41 | \( 1 - 54.1T + 1.68e3T^{2} \) |
| 43 | \( 1 - 35.6iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 64.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 87.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 66.6iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 16.8T + 3.72e3T^{2} \) |
| 67 | \( 1 + 21.2iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 64.2iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 99.4T + 5.32e3T^{2} \) |
| 79 | \( 1 + 139. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 6.03iT - 6.88e3T^{2} \) |
| 89 | \( 1 + 23.9T + 7.92e3T^{2} \) |
| 97 | \( 1 - 171.T + 9.40e3T^{2} \) |
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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
−10.72758632580609977280961562782, −9.284506643396234254469790797196, −8.453853894243900558364788404120, −7.71557554886401109335183399222, −6.57943906230380550354683139474, −6.06951400056150015492659556388, −4.65713578082220318098334902861, −3.07224379908793761701139431391, −1.89346844064763723627466229157, −0.36600560851429686967204483019,
2.09906353488987261176171203065, 3.76595839737794330417719030503, 4.43566595374380455219534884013, 5.33119408615942954894193460212, 6.77283607910587533163057085713, 7.64741811629365531175407654668, 8.939143796348436719362932315250, 9.680177056893601103366903594710, 10.28872491943570267251114600849, 11.12632374555967809490037249650
