| L(s) = 1 | − 2·2-s + 4·4-s + 6·5-s − 8·8-s − 12·10-s − 30·11-s − 2·13-s + 16·16-s + 66·17-s + 52·19-s + 24·20-s + 60·22-s − 114·23-s − 89·25-s + 4·26-s − 72·29-s + 196·31-s − 32·32-s − 132·34-s − 286·37-s − 104·38-s − 48·40-s − 378·41-s + 164·43-s − 120·44-s + 228·46-s − 228·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.536·5-s − 0.353·8-s − 0.379·10-s − 0.822·11-s − 0.0426·13-s + 1/4·16-s + 0.941·17-s + 0.627·19-s + 0.268·20-s + 0.581·22-s − 1.03·23-s − 0.711·25-s + 0.0301·26-s − 0.461·29-s + 1.13·31-s − 0.176·32-s − 0.665·34-s − 1.27·37-s − 0.443·38-s − 0.189·40-s − 1.43·41-s + 0.581·43-s − 0.411·44-s + 0.730·46-s − 0.707·47-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 | \( 1 + p T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - 6 T + p^{3} T^{2} \) |
| 11 | \( 1 + 30 T + p^{3} T^{2} \) |
| 13 | \( 1 + 2 T + p^{3} T^{2} \) |
| 17 | \( 1 - 66 T + p^{3} T^{2} \) |
| 19 | \( 1 - 52 T + p^{3} T^{2} \) |
| 23 | \( 1 + 114 T + p^{3} T^{2} \) |
| 29 | \( 1 + 72 T + p^{3} T^{2} \) |
| 31 | \( 1 - 196 T + p^{3} T^{2} \) |
| 37 | \( 1 + 286 T + p^{3} T^{2} \) |
| 41 | \( 1 + 378 T + p^{3} T^{2} \) |
| 43 | \( 1 - 164 T + p^{3} T^{2} \) |
| 47 | \( 1 + 228 T + p^{3} T^{2} \) |
| 53 | \( 1 - 348 T + p^{3} T^{2} \) |
| 59 | \( 1 + 348 T + p^{3} T^{2} \) |
| 61 | \( 1 - 106 T + p^{3} T^{2} \) |
| 67 | \( 1 - 596 T + p^{3} T^{2} \) |
| 71 | \( 1 + 630 T + p^{3} T^{2} \) |
| 73 | \( 1 - 1042 T + p^{3} T^{2} \) |
| 79 | \( 1 + 88 T + p^{3} T^{2} \) |
| 83 | \( 1 + 1440 T + p^{3} T^{2} \) |
| 89 | \( 1 - 1374 T + p^{3} T^{2} \) |
| 97 | \( 1 - 34 T + p^{3} T^{2} \) |
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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
−9.558998757719835334369672290920, −8.383214222939481173119503532025, −7.82639343617231311496367007621, −6.87414883245019214785990900827, −5.84607362542699487640131407267, −5.15000225408950394502137665612, −3.64332908176095657160530089066, −2.51601187284871032215428564529, −1.43666607632335274257302355314, 0,
1.43666607632335274257302355314, 2.51601187284871032215428564529, 3.64332908176095657160530089066, 5.15000225408950394502137665612, 5.84607362542699487640131407267, 6.87414883245019214785990900827, 7.82639343617231311496367007621, 8.383214222939481173119503532025, 9.558998757719835334369672290920
