| L(s) = 1 | − 33.5i·5-s − 14.6·7-s − 144. i·11-s + 19.3·13-s − 166. i·17-s + 173.·19-s − 220. i·23-s − 503.·25-s − 1.28e3i·29-s + 171.·31-s + 491. i·35-s + 1.26e3·37-s − 1.69e3i·41-s + 878.·43-s + 1.32e3i·47-s + ⋯ |
| L(s) = 1 | − 1.34i·5-s − 0.298·7-s − 1.19i·11-s + 0.114·13-s − 0.574i·17-s + 0.481·19-s − 0.417i·23-s − 0.805·25-s − 1.53i·29-s + 0.178·31-s + 0.401i·35-s + 0.923·37-s − 1.00i·41-s + 0.474·43-s + 0.599i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.410720793\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.410720793\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 33.5iT - 625T^{2} \) |
| 7 | \( 1 + 14.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 144. iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 19.3T + 2.85e4T^{2} \) |
| 17 | \( 1 + 166. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 173.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 220. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 1.28e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 171.T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.26e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 1.69e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 - 878.T + 3.41e6T^{2} \) |
| 47 | \( 1 - 1.32e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 1.67e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 1.73e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.00e3T + 1.38e7T^{2} \) |
| 67 | \( 1 - 7.99e3T + 2.01e7T^{2} \) |
| 71 | \( 1 + 2.79e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 1.21e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 8.78e3T + 3.89e7T^{2} \) |
| 83 | \( 1 + 5.91e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.27e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 6.35e3T + 8.85e7T^{2} \) |
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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
−9.159212926748205613084395988752, −8.388407031621173322325984225984, −7.70504227022327181833142472185, −6.39461187662589353930334038880, −5.62021429152032897532174505569, −4.74779423426692392095700465078, −3.77524100376572840300898269236, −2.57917047113273542713561720573, −1.07705877515617289612103683507, −0.34968454395875616886539293567,
1.51083685432888611097849762222, 2.68595422805370351139040663517, 3.50858921592397713076180995038, 4.63098903575406308383229211411, 5.81745570010001545525654724343, 6.76087507836461137570110816691, 7.24640087929079682418605525242, 8.194695066286752415416560046990, 9.434393542799043806725472306835, 10.00504982285495328030527648231
