| L(s) = 1 | − 5-s − 2·17-s − 4·19-s + 4·23-s + 25-s + 6·29-s + 8·31-s + 6·37-s − 2·41-s − 4·43-s − 7·49-s − 6·53-s − 2·61-s + 8·67-s − 6·73-s + 4·79-s + 12·83-s + 2·85-s + 6·89-s + 4·95-s + 10·97-s + 101-s + 103-s + 107-s + 109-s + 113-s − 4·115-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.485·17-s − 0.917·19-s + 0.834·23-s + 1/5·25-s + 1.11·29-s + 1.43·31-s + 0.986·37-s − 0.312·41-s − 0.609·43-s − 49-s − 0.824·53-s − 0.256·61-s + 0.977·67-s − 0.702·73-s + 0.450·79-s + 1.31·83-s + 0.216·85-s + 0.635·89-s + 0.410·95-s + 1.01·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + 0.0940·113-s − 0.373·115-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 243360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−13.23377038228117, −12.58742332873740, −12.18961172458718, −11.73224180539778, −11.25390288889566, −10.81326505713935, −10.43061548222502, −9.823117228368908, −9.433847972092487, −8.805790759623892, −8.393065580288920, −8.015370122701498, −7.549191258442288, −6.778247053478167, −6.486039421608306, −6.196155666707638, −5.265262774802629, −4.848652185936863, −4.429148991828879, −3.918002121834843, −3.195276709962411, −2.762490603179610, −2.187794580593444, −1.390062716540721, −0.7730429588246797, 0,
0.7730429588246797, 1.390062716540721, 2.187794580593444, 2.762490603179610, 3.195276709962411, 3.918002121834843, 4.429148991828879, 4.848652185936863, 5.265262774802629, 6.196155666707638, 6.486039421608306, 6.778247053478167, 7.549191258442288, 8.015370122701498, 8.393065580288920, 8.805790759623892, 9.433847972092487, 9.823117228368908, 10.43061548222502, 10.81326505713935, 11.25390288889566, 11.73224180539778, 12.18961172458718, 12.58742332873740, 13.23377038228117
