| L(s) = 1 | + 4.85·2-s + 3.92·3-s + 15.5·4-s + 7.28·5-s + 19.0·6-s − 33.6·7-s + 36.5·8-s − 11.5·9-s + 35.3·10-s − 11·11-s + 61.0·12-s − 163.·14-s + 28.6·15-s + 53.2·16-s + 19.6·17-s − 56.1·18-s − 140.·19-s + 113.·20-s − 132.·21-s − 53.3·22-s + 49.5·23-s + 143.·24-s − 71.9·25-s − 151.·27-s − 522.·28-s − 51.1·29-s + 138.·30-s + ⋯ |
| L(s) = 1 | + 1.71·2-s + 0.755·3-s + 1.94·4-s + 0.651·5-s + 1.29·6-s − 1.81·7-s + 1.61·8-s − 0.428·9-s + 1.11·10-s − 0.301·11-s + 1.46·12-s − 3.11·14-s + 0.492·15-s + 0.831·16-s + 0.280·17-s − 0.735·18-s − 1.69·19-s + 1.26·20-s − 1.37·21-s − 0.517·22-s + 0.449·23-s + 1.22·24-s − 0.575·25-s − 1.07·27-s − 3.52·28-s − 0.327·29-s + 0.844·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1859 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 11 | \( 1 + 11T \) |
| 13 | \( 1 \) |
| good | 2 | \( 1 - 4.85T + 8T^{2} \) |
| 3 | \( 1 - 3.92T + 27T^{2} \) |
| 5 | \( 1 - 7.28T + 125T^{2} \) |
| 7 | \( 1 + 33.6T + 343T^{2} \) |
| 17 | \( 1 - 19.6T + 4.91e3T^{2} \) |
| 19 | \( 1 + 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 49.5T + 1.21e4T^{2} \) |
| 29 | \( 1 + 51.1T + 2.43e4T^{2} \) |
| 31 | \( 1 + 28.5T + 2.97e4T^{2} \) |
| 37 | \( 1 + 157.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 253.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 37.2T + 7.95e4T^{2} \) |
| 47 | \( 1 - 579.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 412.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 423.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 249.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 635.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 718.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 1.15e3T + 3.89e5T^{2} \) |
| 79 | \( 1 + 388.T + 4.93e5T^{2} \) |
| 83 | \( 1 + 616.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.05e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.11e3T + 9.12e5T^{2} \) |
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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
−8.585606803856572552039672144696, −7.31190284599908324800697431804, −6.57398895564289668008456436248, −5.94997517312382934606905342331, −5.37650160826345035061269418435, −4.07613073572611803752262068647, −3.47912525211879763783642193432, −2.69277052799655231740459918070, −2.10806371674581315788656654766, 0,
2.10806371674581315788656654766, 2.69277052799655231740459918070, 3.47912525211879763783642193432, 4.07613073572611803752262068647, 5.37650160826345035061269418435, 5.94997517312382934606905342331, 6.57398895564289668008456436248, 7.31190284599908324800697431804, 8.585606803856572552039672144696
