| L(s) = 1 | + 2.82i·5-s + 14.1·7-s − 20i·11-s + 39.5i·13-s + 34·17-s − 52i·19-s − 62.2·23-s + 117·25-s − 200. i·29-s + 110.·31-s + 40.0i·35-s + 271. i·37-s − 26·41-s + 252i·43-s + 345.·47-s + ⋯ |
| L(s) = 1 | + 0.252i·5-s + 0.763·7-s − 0.548i·11-s + 0.844i·13-s + 0.485·17-s − 0.627i·19-s − 0.564·23-s + 0.936·25-s − 1.28i·29-s + 0.639·31-s + 0.193i·35-s + 1.20i·37-s − 0.0990·41-s + 0.893i·43-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1152 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(2.382605150\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.382605150\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - 2.82iT - 125T^{2} \) |
| 7 | \( 1 - 14.1T + 343T^{2} \) |
| 11 | \( 1 + 20iT - 1.33e3T^{2} \) |
| 13 | \( 1 - 39.5iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 34T + 4.91e3T^{2} \) |
| 19 | \( 1 + 52iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 62.2T + 1.21e4T^{2} \) |
| 29 | \( 1 + 200. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 110.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 271. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 26T + 6.89e4T^{2} \) |
| 43 | \( 1 - 252iT - 7.95e4T^{2} \) |
| 47 | \( 1 - 345.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 681. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 364iT - 2.05e5T^{2} \) |
| 61 | \( 1 - 735. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 628iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 338T + 3.89e5T^{2} \) |
| 79 | \( 1 - 789.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.03e3iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 234T + 7.04e5T^{2} \) |
| 97 | \( 1 + 178T + 9.12e5T^{2} \) |
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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
−9.448586314388397858007695431757, −8.472602831773083088207651810425, −7.922493081939238702153151799002, −6.86696134179641662296807276155, −6.14984553361220615793207428036, −5.05105394392340921228318370694, −4.28556533417888526557360441913, −3.12214587073399288990465150740, −2.02914152183182118196556125037, −0.796901520061660001420263519702,
0.847554099171917711023957890501, 1.92922015367171837719337752878, 3.16850331101589513900898739075, 4.28906949436810009805523582003, 5.16276399423517679026567943673, 5.88406683967119816162935653518, 7.10658231920980551392467319245, 7.80056286035743011411088566610, 8.549614019868956931673467460041, 9.360950878554489562601744559365
