| L(s) = 1 | − 10.9·2-s + 26.7·3-s + 87.0·4-s − 91.1·5-s − 291.·6-s − 601.·8-s + 472.·9-s + 994.·10-s + 261.·11-s + 2.32e3·12-s + 169·13-s − 2.43e3·15-s + 3.77e3·16-s − 1.12e3·17-s − 5.15e3·18-s − 33.7·19-s − 7.93e3·20-s − 2.85e3·22-s − 2.25e3·23-s − 1.60e4·24-s + 5.18e3·25-s − 1.84e3·26-s + 6.14e3·27-s + 2.71e3·29-s + 2.66e4·30-s + 6.02e3·31-s − 2.19e4·32-s + ⋯ |
| L(s) = 1 | − 1.92·2-s + 1.71·3-s + 2.72·4-s − 1.63·5-s − 3.31·6-s − 3.32·8-s + 1.94·9-s + 3.14·10-s + 0.652·11-s + 4.67·12-s + 0.277·13-s − 2.79·15-s + 3.68·16-s − 0.946·17-s − 3.75·18-s − 0.0214·19-s − 4.43·20-s − 1.25·22-s − 0.889·23-s − 5.69·24-s + 1.65·25-s − 0.535·26-s + 1.62·27-s + 0.598·29-s + 5.39·30-s + 1.12·31-s − 3.78·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 7 | \( 1 \) |
| 13 | \( 1 - 169T \) |
| good | 2 | \( 1 + 10.9T + 32T^{2} \) |
| 3 | \( 1 - 26.7T + 243T^{2} \) |
| 5 | \( 1 + 91.1T + 3.12e3T^{2} \) |
| 11 | \( 1 - 261.T + 1.61e5T^{2} \) |
| 17 | \( 1 + 1.12e3T + 1.41e6T^{2} \) |
| 19 | \( 1 + 33.7T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.25e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 6.02e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.63e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.31e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.07e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 4.06e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.66e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 3.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 4.30e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 2.94e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 3.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 9.80e3T + 3.07e9T^{2} \) |
| 83 | \( 1 - 2.35e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 2.51e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 1.47e5T + 8.58e9T^{2} \) |
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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
−9.030700635570960590860379099635, −8.410608633951118782471272457642, −8.123459410113428906910928457499, −7.21597850551007372283374373541, −6.62058920509043310297538008242, −4.16346699279894952424533779517, −3.30980146733851177331499082905, −2.34305591953969115343745471817, −1.22871156048873169821877961469, 0,
1.22871156048873169821877961469, 2.34305591953969115343745471817, 3.30980146733851177331499082905, 4.16346699279894952424533779517, 6.62058920509043310297538008242, 7.21597850551007372283374373541, 8.123459410113428906910928457499, 8.410608633951118782471272457642, 9.030700635570960590860379099635
