| L(s) = 1 | + 4·2-s + 9.04·3-s + 16·4-s + 36.1·6-s − 77.0·7-s + 64·8-s − 161.·9-s + 463.·11-s + 144.·12-s + 169·13-s − 308.·14-s + 256·16-s − 1.86e3·17-s − 644.·18-s − 618.·19-s − 696.·21-s + 1.85e3·22-s + 1.71e3·23-s + 578.·24-s + 676·26-s − 3.65e3·27-s − 1.23e3·28-s − 5.86e3·29-s − 545.·31-s + 1.02e3·32-s + 4.18e3·33-s − 7.47e3·34-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.580·3-s + 0.5·4-s + 0.410·6-s − 0.594·7-s + 0.353·8-s − 0.663·9-s + 1.15·11-s + 0.290·12-s + 0.277·13-s − 0.420·14-s + 0.250·16-s − 1.56·17-s − 0.469·18-s − 0.392·19-s − 0.344·21-s + 0.816·22-s + 0.674·23-s + 0.205·24-s + 0.196·26-s − 0.964·27-s − 0.297·28-s − 1.29·29-s − 0.101·31-s + 0.176·32-s + 0.669·33-s − 1.10·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 650 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - 4T \) |
| 5 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 3 | \( 1 - 9.04T + 243T^{2} \) |
| 7 | \( 1 + 77.0T + 1.68e4T^{2} \) |
| 11 | \( 1 - 463.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.86e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 618.T + 2.47e6T^{2} \) |
| 23 | \( 1 - 1.71e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.86e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 545.T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.23e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.77e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.26e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.59e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.79e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 5.54e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 5.96e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 6.72e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.73e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.90e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.25e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.36e3T + 5.58e9T^{2} \) |
| 97 | \( 1 + 5.80e4T + 8.58e9T^{2} \) |
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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
−9.044082908047256219290780469585, −8.768262404362406948643589173127, −7.40729381465700965619635706651, −6.53373015010517048251222108015, −5.84359426597826565784655723177, −4.51453078887353720969381314104, −3.66171562468831210332161180985, −2.76672718956266314142823568825, −1.67494515581758385735366203790, 0,
1.67494515581758385735366203790, 2.76672718956266314142823568825, 3.66171562468831210332161180985, 4.51453078887353720969381314104, 5.84359426597826565784655723177, 6.53373015010517048251222108015, 7.40729381465700965619635706651, 8.768262404362406948643589173127, 9.044082908047256219290780469585
