| L(s) = 1 | + 2-s − 4-s − 2·5-s − 7-s − 3·8-s − 2·10-s + 11-s − 2·13-s − 14-s − 16-s − 2·17-s + 6·19-s + 2·20-s + 22-s − 4·23-s − 25-s − 2·26-s + 28-s + 29-s + 4·31-s + 5·32-s − 2·34-s + 2·35-s + 6·37-s + 6·38-s + 6·40-s − 2·41-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s − 0.377·7-s − 1.06·8-s − 0.632·10-s + 0.301·11-s − 0.554·13-s − 0.267·14-s − 1/4·16-s − 0.485·17-s + 1.37·19-s + 0.447·20-s + 0.213·22-s − 0.834·23-s − 1/5·25-s − 0.392·26-s + 0.188·28-s + 0.185·29-s + 0.718·31-s + 0.883·32-s − 0.342·34-s + 0.338·35-s + 0.986·37-s + 0.973·38-s + 0.948·40-s − 0.312·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20097 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20097 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 12 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 14 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 16 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 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
−15.97970056877609, −15.21658812991508, −14.88938608280919, −14.23555514772392, −13.69078376510144, −13.33213053579498, −12.63035171669853, −12.06760791573727, −11.71825350982417, −11.31902244578379, −10.21241705702864, −9.841205837674909, −9.266277440611871, −8.595508551683813, −8.003443846277041, −7.460057095318397, −6.703957734430274, −6.103813885454724, −5.415956064925427, −4.722503097863365, −4.264411097626966, −3.533043193561677, −3.118756505931404, −2.178663582128612, −0.8752436233993046, 0,
0.8752436233993046, 2.178663582128612, 3.118756505931404, 3.533043193561677, 4.264411097626966, 4.722503097863365, 5.415956064925427, 6.103813885454724, 6.703957734430274, 7.460057095318397, 8.003443846277041, 8.595508551683813, 9.266277440611871, 9.841205837674909, 10.21241705702864, 11.31902244578379, 11.71825350982417, 12.06760791573727, 12.63035171669853, 13.33213053579498, 13.69078376510144, 14.23555514772392, 14.88938608280919, 15.21658812991508, 15.97970056877609
