| L(s) = 1 | − 2·3-s − 5-s + 9-s − 2·11-s + 2·15-s + 2·17-s + 4·19-s + 4·23-s + 25-s + 4·27-s − 4·29-s + 31-s + 4·33-s − 8·37-s + 6·41-s − 2·43-s − 45-s − 7·49-s − 4·51-s + 8·53-s + 2·55-s − 8·57-s − 8·59-s − 4·67-s − 8·69-s + 6·73-s − 2·75-s + ⋯ |
| L(s) = 1 | − 1.15·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s + 0.516·15-s + 0.485·17-s + 0.917·19-s + 0.834·23-s + 1/5·25-s + 0.769·27-s − 0.742·29-s + 0.179·31-s + 0.696·33-s − 1.31·37-s + 0.937·41-s − 0.304·43-s − 0.149·45-s − 49-s − 0.560·51-s + 1.09·53-s + 0.269·55-s − 1.05·57-s − 1.04·59-s − 0.488·67-s − 0.963·69-s + 0.702·73-s − 0.230·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2480 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| good | 3 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 8 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.492991502396247670022058615050, −7.62719225217277156404644359290, −7.01969100272950224003714185366, −6.10742676075883314085892542646, −5.33072080167810451628709907496, −4.87810182943580631405319775503, −3.69683871468629610845615643427, −2.77418319730261383273863638001, −1.22167689552642552772774605158, 0,
1.22167689552642552772774605158, 2.77418319730261383273863638001, 3.69683871468629610845615643427, 4.87810182943580631405319775503, 5.33072080167810451628709907496, 6.10742676075883314085892542646, 7.01969100272950224003714185366, 7.62719225217277156404644359290, 8.492991502396247670022058615050
