| L(s) = 1 | − 2-s + 4-s + 5-s + 2·7-s − 8-s − 10-s + 4·11-s − 2·14-s + 16-s − 4·17-s + 2·19-s + 20-s − 4·22-s − 2·23-s + 25-s + 2·28-s − 8·29-s − 4·31-s − 32-s + 4·34-s + 2·35-s − 6·37-s − 2·38-s − 40-s + 10·41-s + 4·43-s + 4·44-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 + 1.20·11-s − 0.534·14-s + 1/4·16-s − 0.970·17-s + 0.458·19-s + 0.223·20-s − 0.852·22-s − 0.417·23-s + 1/5·25-s + 0.377·28-s − 1.48·29-s − 0.718·31-s − 0.176·32-s + 0.685·34-s + 0.338·35-s − 0.986·37-s − 0.324·38-s − 0.158·40-s + 1.56·41-s + 0.609·43-s + 0.603·44-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 15210 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 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
−16.38308700951415, −15.86615201277687, −15.21997762706835, −14.63569240002305, −14.15278338268653, −13.73222626170028, −12.78435033701438, −12.42553206578810, −11.57631028300286, −11.11908273920503, −10.86631211654989, −9.866311722283144, −9.417311651958307, −8.950387280734154, −8.408783442653696, −7.552834637200706, −7.197868491715419, −6.360529590456373, −5.876770429578080, −5.100515822305415, −4.275999506041381, −3.642731625436056, −2.612811258263634, −1.771159711570785, −1.330354154240752, 0,
1.330354154240752, 1.771159711570785, 2.612811258263634, 3.642731625436056, 4.275999506041381, 5.100515822305415, 5.876770429578080, 6.360529590456373, 7.197868491715419, 7.552834637200706, 8.408783442653696, 8.950387280734154, 9.417311651958307, 9.866311722283144, 10.86631211654989, 11.11908273920503, 11.57631028300286, 12.42553206578810, 12.78435033701438, 13.73222626170028, 14.15278338268653, 14.63569240002305, 15.21997762706835, 15.86615201277687, 16.38308700951415
