| L(s) = 1 | − 2-s + 3·3-s + 4-s + 2·5-s − 3·6-s + 4·7-s − 8-s + 6·9-s − 2·10-s − 2·11-s + 3·12-s + 13-s − 4·14-s + 6·15-s + 16-s + 17-s − 6·18-s + 2·20-s + 12·21-s + 2·22-s − 23-s − 3·24-s − 25-s − 26-s + 9·27-s + 4·28-s − 3·29-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1.73·3-s + 1/2·4-s + 0.894·5-s − 1.22·6-s + 1.51·7-s − 0.353·8-s + 2·9-s − 0.632·10-s − 0.603·11-s + 0.866·12-s + 0.277·13-s − 1.06·14-s + 1.54·15-s + 1/4·16-s + 0.242·17-s − 1.41·18-s + 0.447·20-s + 2.61·21-s + 0.426·22-s − 0.208·23-s − 0.612·24-s − 1/5·25-s − 0.196·26-s + 1.73·27-s + 0.755·28-s − 0.557·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2006 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.271858307\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.271858307\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + T \) |
| 17 | \( 1 - T \) |
| 59 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 11 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 4 T + p T^{2} \) |
| 97 | \( 1 - 17 T + p T^{2} \) |
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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
−9.025092674509700560877384246685, −8.465244029576558181564598805292, −7.71111047057350297077502374079, −7.42023326789824554033466062772, −6.05866257668836816509118980785, −5.11197903780870040710661682109, −4.03994524398527615534761270756, −2.93246722212421787890182993312, −2.01369154116906829839844478556, −1.53772835375377315132012235922,
1.53772835375377315132012235922, 2.01369154116906829839844478556, 2.93246722212421787890182993312, 4.03994524398527615534761270756, 5.11197903780870040710661682109, 6.05866257668836816509118980785, 7.42023326789824554033466062772, 7.71111047057350297077502374079, 8.465244029576558181564598805292, 9.025092674509700560877384246685
