| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s − 2·14-s + 16-s + 17-s + 20-s + 4·23-s + 25-s + 2·28-s + 2·29-s + 4·31-s − 32-s − 34-s + 2·35-s − 2·37-s − 40-s + 6·43-s − 4·46-s − 3·49-s − 50-s + 6·53-s − 2·56-s − 2·58-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s + 0.755·7-s − 0.353·8-s − 0.316·10-s − 0.534·14-s + 1/4·16-s + 0.242·17-s + 0.223·20-s + 0.834·23-s + 1/5·25-s + 0.377·28-s + 0.371·29-s + 0.718·31-s − 0.176·32-s − 0.171·34-s + 0.338·35-s − 0.328·37-s − 0.158·40-s + 0.914·43-s − 0.589·46-s − 3/7·49-s − 0.141·50-s + 0.824·53-s − 0.267·56-s − 0.262·58-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 185130 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 + 16 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 18 T + p T^{2} \) |
| 97 | \( 1 + p T^{2} \) |
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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
−13.37599290377366, −12.90012116723018, −12.26428539522326, −11.96343207836127, −11.35160039931334, −10.98482713117657, −10.47172274619837, −10.13700021199591, −9.508495033930662, −9.088715661811337, −8.688214278313687, −8.050005573035561, −7.800479027234388, −7.147535770683359, −6.657771108595105, −6.216442765909467, −5.485071067460911, −5.194588835360795, −4.497209669450160, −3.993031324831889, −3.114082418504866, −2.706972506052365, −2.059177784380155, −1.358485938246234, −0.9895782058451671, 0,
0.9895782058451671, 1.358485938246234, 2.059177784380155, 2.706972506052365, 3.114082418504866, 3.993031324831889, 4.497209669450160, 5.194588835360795, 5.485071067460911, 6.216442765909467, 6.657771108595105, 7.147535770683359, 7.800479027234388, 8.050005573035561, 8.688214278313687, 9.088715661811337, 9.508495033930662, 10.13700021199591, 10.47172274619837, 10.98482713117657, 11.35160039931334, 11.96343207836127, 12.26428539522326, 12.90012116723018, 13.37599290377366
