| L(s) = 1 | + 4.19·2-s + 4.70·3-s + 9.56·4-s + 14.4·5-s + 19.7·6-s − 34.7·7-s + 6.55·8-s − 4.85·9-s + 60.4·10-s + 11·11-s + 44.9·12-s − 145.·14-s + 67.8·15-s − 49.0·16-s − 75.2·17-s − 20.3·18-s + 50.5·19-s + 137.·20-s − 163.·21-s + 46.0·22-s − 170.·23-s + 30.8·24-s + 83.1·25-s − 149.·27-s − 331.·28-s − 265.·29-s + 284.·30-s + ⋯ |
| L(s) = 1 | + 1.48·2-s + 0.905·3-s + 1.19·4-s + 1.29·5-s + 1.34·6-s − 1.87·7-s + 0.289·8-s − 0.179·9-s + 1.91·10-s + 0.301·11-s + 1.08·12-s − 2.77·14-s + 1.16·15-s − 0.766·16-s − 1.07·17-s − 0.266·18-s + 0.610·19-s + 1.54·20-s − 1.69·21-s + 0.446·22-s − 1.54·23-s + 0.262·24-s + 0.664·25-s − 1.06·27-s − 2.23·28-s − 1.70·29-s + 1.73·30-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 | \( 1 - 4.19T + 8T^{2} \) |
| 3 | \( 1 - 4.70T + 27T^{2} \) |
| 5 | \( 1 - 14.4T + 125T^{2} \) |
| 7 | \( 1 + 34.7T + 343T^{2} \) |
| 17 | \( 1 + 75.2T + 4.91e3T^{2} \) |
| 19 | \( 1 - 50.5T + 6.85e3T^{2} \) |
| 23 | \( 1 + 170.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 265.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 134.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 7.36T + 5.06e4T^{2} \) |
| 41 | \( 1 - 106.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 437.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 374.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 650.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 278.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 52.5T + 2.26e5T^{2} \) |
| 67 | \( 1 + 85.0T + 3.00e5T^{2} \) |
| 71 | \( 1 - 264.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 269.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 214.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.12e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 873.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 520.T + 9.12e5T^{2} \) |
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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
−8.754592017969751120615579754843, −7.36516447918278845364699436960, −6.54112505520639903238324779650, −5.90312472444625853994872425996, −5.49909633609640558490557746372, −3.98036910699759177549040237589, −3.56516093347108948537745177128, −2.55329243188958417666010129370, −2.11311043645931395233237110935, 0,
2.11311043645931395233237110935, 2.55329243188958417666010129370, 3.56516093347108948537745177128, 3.98036910699759177549040237589, 5.49909633609640558490557746372, 5.90312472444625853994872425996, 6.54112505520639903238324779650, 7.36516447918278845364699436960, 8.754592017969751120615579754843
