| L(s) = 1 | − 5-s − 11-s − 2·13-s − 2·17-s − 4·19-s − 6·23-s + 25-s − 10·29-s − 2·31-s + 2·37-s + 10·41-s − 2·43-s − 8·47-s − 2·53-s + 55-s + 4·59-s + 2·61-s + 2·65-s + 4·67-s − 6·71-s + 2·73-s + 8·79-s − 6·83-s + 2·85-s − 10·89-s + 4·95-s − 8·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s − 0.554·13-s − 0.485·17-s − 0.917·19-s − 1.25·23-s + 1/5·25-s − 1.85·29-s − 0.359·31-s + 0.328·37-s + 1.56·41-s − 0.304·43-s − 1.16·47-s − 0.274·53-s + 0.134·55-s + 0.520·59-s + 0.256·61-s + 0.248·65-s + 0.488·67-s − 0.712·71-s + 0.234·73-s + 0.900·79-s − 0.658·83-s + 0.216·85-s − 1.05·89-s + 0.410·95-s − 0.812·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 388080 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−12.79249502918133, −12.25759813352385, −11.73903058802417, −11.28597134562568, −10.98672263887083, −10.47910573799779, −9.979098307346721, −9.433170161625683, −9.215566834384277, −8.486317308245375, −8.047971790687612, −7.810731592197274, −7.075671466374857, −6.915757245336289, −6.067452699555209, −5.849729526954514, −5.221260363886017, −4.645757093389480, −4.166087462298798, −3.808689819258724, −3.186752293739867, −2.474582879202470, −2.078366521988588, −1.533073536352068, −0.5157020030925732, 0,
0.5157020030925732, 1.533073536352068, 2.078366521988588, 2.474582879202470, 3.186752293739867, 3.808689819258724, 4.166087462298798, 4.645757093389480, 5.221260363886017, 5.849729526954514, 6.067452699555209, 6.915757245336289, 7.075671466374857, 7.810731592197274, 8.047971790687612, 8.486317308245375, 9.215566834384277, 9.433170161625683, 9.979098307346721, 10.47910573799779, 10.98672263887083, 11.28597134562568, 11.73903058802417, 12.25759813352385, 12.79249502918133
