| L(s) = 1 | − 4·2-s + 16·4-s − 2.39·5-s − 51.6·7-s − 64·8-s + 9.57·10-s + 670.·11-s − 846.·13-s + 206.·14-s + 256·16-s − 1.13e3·17-s + 1.00e3·19-s − 38.3·20-s − 2.68e3·22-s + 2.40e3·23-s − 3.11e3·25-s + 3.38e3·26-s − 827.·28-s + 3.76e3·29-s + 4.03e3·31-s − 1.02e3·32-s + 4.54e3·34-s + 123.·35-s − 1.45e4·37-s − 4.00e3·38-s + 153.·40-s − 7.88e3·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s − 0.0428·5-s − 0.398·7-s − 0.353·8-s + 0.0302·10-s + 1.67·11-s − 1.38·13-s + 0.281·14-s + 0.250·16-s − 0.952·17-s + 0.637·19-s − 0.0214·20-s − 1.18·22-s + 0.946·23-s − 0.998·25-s + 0.982·26-s − 0.199·28-s + 0.832·29-s + 0.753·31-s − 0.176·32-s + 0.673·34-s + 0.0170·35-s − 1.74·37-s − 0.450·38-s + 0.0151·40-s − 0.732·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 4T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 2.39T + 3.12e3T^{2} \) |
| 7 | \( 1 + 51.6T + 1.68e4T^{2} \) |
| 11 | \( 1 - 670.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 846.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.13e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.00e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.40e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 3.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 4.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.45e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 7.88e3T + 1.15e8T^{2} \) |
| 43 | \( 1 + 2.03e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.61e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 5.54e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 6.41e3T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.88e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.29e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.33e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 1.28e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 2.92e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.11e5T + 8.58e9T^{2} \) |
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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
−11.61298425393125072477097267809, −10.18230866603900699321846591041, −9.442414125922546825429919912384, −8.540636407442362719359062707768, −7.13806529114169311531517972716, −6.46618867659138973536813763792, −4.78352265788203075760844385039, −3.21747633950733596213687698259, −1.62572664444521784498626883956, 0,
1.62572664444521784498626883956, 3.21747633950733596213687698259, 4.78352265788203075760844385039, 6.46618867659138973536813763792, 7.13806529114169311531517972716, 8.540636407442362719359062707768, 9.442414125922546825429919912384, 10.18230866603900699321846591041, 11.61298425393125072477097267809
