| L(s) = 1 | − 2-s − 4-s + 5-s − 4·7-s + 3·8-s − 3·9-s − 10-s − 2·13-s + 4·14-s − 16-s − 6·17-s + 3·18-s − 20-s − 8·23-s + 25-s + 2·26-s + 4·28-s + 6·29-s − 5·32-s + 6·34-s − 4·35-s + 3·36-s + 6·37-s + 3·40-s + 41-s + 4·43-s − 3·45-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 0.447·5-s − 1.51·7-s + 1.06·8-s − 9-s − 0.316·10-s − 0.554·13-s + 1.06·14-s − 1/4·16-s − 1.45·17-s + 0.707·18-s − 0.223·20-s − 1.66·23-s + 1/5·25-s + 0.392·26-s + 0.755·28-s + 1.11·29-s − 0.883·32-s + 1.02·34-s − 0.676·35-s + 1/2·36-s + 0.986·37-s + 0.474·40-s + 0.156·41-s + 0.609·43-s − 0.447·45-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 205 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 5 | \( 1 - T \) |
| 41 | \( 1 - T \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 6 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−11.93450768172631259062214800288, −10.59674722336712637467004889245, −9.786822083448618856097399512400, −9.104973006323854487272781038861, −8.185899683953959523799523920888, −6.77337413472352473188697214294, −5.78635720535852048108327814868, −4.25020877925447943629142282453, −2.60807987098564229857754881826, 0,
2.60807987098564229857754881826, 4.25020877925447943629142282453, 5.78635720535852048108327814868, 6.77337413472352473188697214294, 8.185899683953959523799523920888, 9.104973006323854487272781038861, 9.786822083448618856097399512400, 10.59674722336712637467004889245, 11.93450768172631259062214800288
