| L(s) = 1 | − 2·2-s + 4·4-s + 4.58·5-s − 8·8-s − 9.17·10-s + 6.48·11-s − 45.2·13-s + 16·16-s + 81.5·17-s + 5.05·19-s + 18.3·20-s − 12.9·22-s − 106.·23-s − 103.·25-s + 90.4·26-s + 268.·29-s − 292.·31-s − 32·32-s − 163.·34-s + 114.·37-s − 10.1·38-s − 36.6·40-s − 161.·41-s − 471.·43-s + 25.9·44-s + 212.·46-s + 346.·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.410·5-s − 0.353·8-s − 0.290·10-s + 0.177·11-s − 0.964·13-s + 0.250·16-s + 1.16·17-s + 0.0610·19-s + 0.205·20-s − 0.125·22-s − 0.963·23-s − 0.831·25-s + 0.682·26-s + 1.71·29-s − 1.69·31-s − 0.176·32-s − 0.822·34-s + 0.509·37-s − 0.0431·38-s − 0.145·40-s − 0.615·41-s − 1.67·43-s + 0.0888·44-s + 0.681·46-s + 1.07·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 4.58T + 125T^{2} \) |
| 11 | \( 1 - 6.48T + 1.33e3T^{2} \) |
| 13 | \( 1 + 45.2T + 2.19e3T^{2} \) |
| 17 | \( 1 - 81.5T + 4.91e3T^{2} \) |
| 19 | \( 1 - 5.05T + 6.85e3T^{2} \) |
| 23 | \( 1 + 106.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 268.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 292.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 114.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 161.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 471.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 346.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 405.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 253.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 751.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 11.6T + 3.00e5T^{2} \) |
| 71 | \( 1 - 681.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 685.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 0.264T + 4.93e5T^{2} \) |
| 83 | \( 1 + 437.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 58.5T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.28e3T + 9.12e5T^{2} \) |
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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
−9.599565989586647906717744167688, −8.451148238818894511191537357510, −7.74620278928510678559336029065, −6.88483501170140171596040817226, −5.92826087059954899252565624188, −5.05225765678717653038698737627, −3.69210546625416640933482738414, −2.50146362503905051724246523165, −1.43187017613334314839745591501, 0,
1.43187017613334314839745591501, 2.50146362503905051724246523165, 3.69210546625416640933482738414, 5.05225765678717653038698737627, 5.92826087059954899252565624188, 6.88483501170140171596040817226, 7.74620278928510678559336029065, 8.451148238818894511191537357510, 9.599565989586647906717744167688
