| L(s) = 1 | − 5.62·2-s + 23.6·4-s − 7.12·5-s − 7·7-s − 87.8·8-s + 40.0·10-s + 11·11-s − 28.1·13-s + 39.3·14-s + 305.·16-s − 53.4·17-s + 154.·19-s − 168.·20-s − 61.8·22-s + 44.4·23-s − 74.3·25-s + 158.·26-s − 165.·28-s + 154.·29-s + 63.2·31-s − 1.01e3·32-s + 300.·34-s + 49.8·35-s − 137.·37-s − 866.·38-s + 625.·40-s − 442.·41-s + ⋯ |
| L(s) = 1 | − 1.98·2-s + 2.95·4-s − 0.636·5-s − 0.377·7-s − 3.88·8-s + 1.26·10-s + 0.301·11-s − 0.600·13-s + 0.751·14-s + 4.76·16-s − 0.762·17-s + 1.86·19-s − 1.88·20-s − 0.599·22-s + 0.403·23-s − 0.594·25-s + 1.19·26-s − 1.11·28-s + 0.988·29-s + 0.366·31-s − 5.59·32-s + 1.51·34-s + 0.240·35-s − 0.611·37-s − 3.69·38-s + 2.47·40-s − 1.68·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 5.62T + 8T^{2} \) |
| 5 | \( 1 + 7.12T + 125T^{2} \) |
| 13 | \( 1 + 28.1T + 2.19e3T^{2} \) |
| 17 | \( 1 + 53.4T + 4.91e3T^{2} \) |
| 19 | \( 1 - 154.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 44.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 154.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 63.2T + 2.97e4T^{2} \) |
| 37 | \( 1 + 137.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 442.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 48.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 122.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 298.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 694.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 7.85T + 2.26e5T^{2} \) |
| 67 | \( 1 - 467.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 984.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 128.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 1.33e3T + 4.93e5T^{2} \) |
| 83 | \( 1 - 984.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 918.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.53e3T + 9.12e5T^{2} \) |
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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
−9.687046461794374040974517545338, −8.761930441574296621735906105898, −8.085987287474218385717142510177, −7.14911389672617409198250646391, −6.71876740315079389585767362723, −5.39809819897996569740280924410, −3.52134576320278526122207731924, −2.49049010650185843913023122691, −1.11533686454598848017752590049, 0,
1.11533686454598848017752590049, 2.49049010650185843913023122691, 3.52134576320278526122207731924, 5.39809819897996569740280924410, 6.71876740315079389585767362723, 7.14911389672617409198250646391, 8.085987287474218385717142510177, 8.761930441574296621735906105898, 9.687046461794374040974517545338
