| L(s) = 1 | − 5.52·2-s − 3·3-s + 22.5·4-s + 16.5·6-s − 22.7·7-s − 80.3·8-s + 9·9-s − 11·11-s − 67.6·12-s − 25.8·13-s + 125.·14-s + 263.·16-s + 103.·17-s − 49.7·18-s − 91.5·19-s + 68.1·21-s + 60.7·22-s + 78.7·23-s + 241.·24-s + 143.·26-s − 27·27-s − 512.·28-s + 243.·29-s − 177.·31-s − 815.·32-s + 33·33-s − 570.·34-s + ⋯ |
| L(s) = 1 | − 1.95·2-s − 0.577·3-s + 2.81·4-s + 1.12·6-s − 1.22·7-s − 3.55·8-s + 0.333·9-s − 0.301·11-s − 1.62·12-s − 0.552·13-s + 2.39·14-s + 4.12·16-s + 1.47·17-s − 0.651·18-s − 1.10·19-s + 0.708·21-s + 0.589·22-s + 0.714·23-s + 2.05·24-s + 1.07·26-s − 0.192·27-s − 3.45·28-s + 1.55·29-s − 1.02·31-s − 4.50·32-s + 0.174·33-s − 2.87·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 + 3T \) |
| 5 | \( 1 \) |
| 11 | \( 1 + 11T \) |
| good | 2 | \( 1 + 5.52T + 8T^{2} \) |
| 7 | \( 1 + 22.7T + 343T^{2} \) |
| 13 | \( 1 + 25.8T + 2.19e3T^{2} \) |
| 17 | \( 1 - 103.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 91.5T + 6.85e3T^{2} \) |
| 23 | \( 1 - 78.7T + 1.21e4T^{2} \) |
| 29 | \( 1 - 243.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 177.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 71.7T + 5.06e4T^{2} \) |
| 41 | \( 1 + 321.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 64.4T + 7.95e4T^{2} \) |
| 47 | \( 1 - 76.7T + 1.03e5T^{2} \) |
| 53 | \( 1 - 181.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 623.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 86.9T + 2.26e5T^{2} \) |
| 67 | \( 1 - 162.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 326.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 728.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 261.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 1.30e3T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.22e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.30e3T + 9.12e5T^{2} \) |
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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
−9.554447083822788210447183615741, −8.688819739695645765491253643530, −7.80238449665437705853474590491, −6.92741972367795369953207225045, −6.38795554235234385262666759949, −5.38384203529774653636577678066, −3.40699797392774084450194159932, −2.38889557093119844929141531671, −0.965895632027186204925227737473, 0,
0.965895632027186204925227737473, 2.38889557093119844929141531671, 3.40699797392774084450194159932, 5.38384203529774653636577678066, 6.38795554235234385262666759949, 6.92741972367795369953207225045, 7.80238449665437705853474590491, 8.688819739695645765491253643530, 9.554447083822788210447183615741
