| L(s) = 1 | + 5·5-s − 10.2·7-s + 5.26·11-s + 34.5·13-s − 74.8·17-s − 50.8·19-s + 81.4·23-s + 25·25-s − 152.·29-s + 93.9·31-s − 51.3·35-s − 167.·37-s − 18.6·41-s + 133.·43-s − 46.8·47-s − 237.·49-s + 105.·53-s + 26.3·55-s + 77.7·59-s − 48.6·61-s + 172.·65-s − 667.·67-s + 344.·71-s − 1.07e3·73-s − 54·77-s − 420.·79-s + 296.·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 0.554·7-s + 0.144·11-s + 0.737·13-s − 1.06·17-s − 0.614·19-s + 0.738·23-s + 0.200·25-s − 0.975·29-s + 0.544·31-s − 0.247·35-s − 0.743·37-s − 0.0709·41-s + 0.473·43-s − 0.145·47-s − 0.692·49-s + 0.272·53-s + 0.0645·55-s + 0.171·59-s − 0.102·61-s + 0.329·65-s − 1.21·67-s + 0.575·71-s − 1.72·73-s − 0.0799·77-s − 0.599·79-s + 0.392·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1080 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5T \) |
| good | 7 | \( 1 + 10.2T + 343T^{2} \) |
| 11 | \( 1 - 5.26T + 1.33e3T^{2} \) |
| 13 | \( 1 - 34.5T + 2.19e3T^{2} \) |
| 17 | \( 1 + 74.8T + 4.91e3T^{2} \) |
| 19 | \( 1 + 50.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 81.4T + 1.21e4T^{2} \) |
| 29 | \( 1 + 152.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 93.9T + 2.97e4T^{2} \) |
| 37 | \( 1 + 167.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 18.6T + 6.89e4T^{2} \) |
| 43 | \( 1 - 133.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 46.8T + 1.03e5T^{2} \) |
| 53 | \( 1 - 105.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 77.7T + 2.05e5T^{2} \) |
| 61 | \( 1 + 48.6T + 2.26e5T^{2} \) |
| 67 | \( 1 + 667.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 344.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.07e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 420.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 296.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 950.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 135.T + 9.12e5T^{2} \) |
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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
−9.032199649222336861415918247741, −8.502953711254384300446765570932, −7.26364445451312869567025546360, −6.50202631623059677650968196447, −5.81659220612376088911722405013, −4.69958410563187524404989879724, −3.71000585339739398128885884258, −2.62849769895208670965553358323, −1.47042105000861003420020340199, 0,
1.47042105000861003420020340199, 2.62849769895208670965553358323, 3.71000585339739398128885884258, 4.69958410563187524404989879724, 5.81659220612376088911722405013, 6.50202631623059677650968196447, 7.26364445451312869567025546360, 8.502953711254384300446765570932, 9.032199649222336861415918247741
