| L(s) = 1 | − 3·3-s − 8.70·5-s + 15.4·7-s + 9·9-s − 11·11-s + 0.704·13-s + 26.1·15-s + 24.1·17-s − 66.1·19-s − 46.2·21-s + 46.4·23-s − 49.2·25-s − 27·27-s + 25.0·29-s + 117.·31-s + 33·33-s − 134.·35-s + 123.·37-s − 2.11·39-s − 282.·41-s − 104.·43-s − 78.3·45-s + 436.·47-s − 105.·49-s − 72.3·51-s + 340.·53-s + 95.7·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.778·5-s + 0.832·7-s + 0.333·9-s − 0.301·11-s + 0.0150·13-s + 0.449·15-s + 0.344·17-s − 0.798·19-s − 0.480·21-s + 0.421·23-s − 0.393·25-s − 0.192·27-s + 0.160·29-s + 0.678·31-s + 0.174·33-s − 0.647·35-s + 0.548·37-s − 0.00868·39-s − 1.07·41-s − 0.368·43-s − 0.259·45-s + 1.35·47-s − 0.307·49-s − 0.198·51-s + 0.882·53-s + 0.234·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2112 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| 3 | \( 1 + 3T \) |
| 11 | \( 1 + 11T \) |
| good | 5 | \( 1 + 8.70T + 125T^{2} \) |
| 7 | \( 1 - 15.4T + 343T^{2} \) |
| 13 | \( 1 - 0.704T + 2.19e3T^{2} \) |
| 17 | \( 1 - 24.1T + 4.91e3T^{2} \) |
| 19 | \( 1 + 66.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 46.4T + 1.21e4T^{2} \) |
| 29 | \( 1 - 25.0T + 2.43e4T^{2} \) |
| 31 | \( 1 - 117.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 123.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 282.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 104.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 436.T + 1.03e5T^{2} \) |
| 53 | \( 1 - 340.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 518.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 466.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 321.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 763.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 644.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 453.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 658.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 466.T + 9.12e5T^{2} \) |
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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
−8.170175762972655215167891453312, −7.70402469988888383477242548892, −6.82053636505028428230313031556, −5.95775631925830384795623087339, −5.02957630687806871130984165753, −4.42302759315703619154937400650, −3.51254979866984090404094790285, −2.26289862778333520701665699965, −1.10031837398796014371417266506, 0,
1.10031837398796014371417266506, 2.26289862778333520701665699965, 3.51254979866984090404094790285, 4.42302759315703619154937400650, 5.02957630687806871130984165753, 5.95775631925830384795623087339, 6.82053636505028428230313031556, 7.70402469988888383477242548892, 8.170175762972655215167891453312
