| L(s) = 1 | − 2-s − 3-s + 4-s + 2·5-s + 6-s − 7-s − 8-s + 9-s − 2·10-s + 4·11-s − 12-s + 14-s − 2·15-s + 16-s − 6·17-s − 18-s − 4·19-s + 2·20-s + 21-s − 4·22-s − 23-s + 24-s − 25-s − 27-s − 28-s − 2·29-s + 2·30-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.894·5-s + 0.408·6-s − 0.377·7-s − 0.353·8-s + 1/3·9-s − 0.632·10-s + 1.20·11-s − 0.288·12-s + 0.267·14-s − 0.516·15-s + 1/4·16-s − 1.45·17-s − 0.235·18-s − 0.917·19-s + 0.447·20-s + 0.218·21-s − 0.852·22-s − 0.208·23-s + 0.204·24-s − 1/5·25-s − 0.192·27-s − 0.188·28-s − 0.371·29-s + 0.365·30-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 163254 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 + T \) |
| 13 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−13.52935188132935, −13.00014249451384, −12.41344512942106, −12.00841213637971, −11.60825323699230, −10.87702185701595, −10.74734079937815, −10.08308492450342, −9.721213247187618, −9.106974467899618, −8.934265028358095, −8.358067540696352, −7.649108491797431, −7.027072003377404, −6.571955517078663, −6.233967830205165, −5.941350783971278, −5.174352323363938, −4.490886974829638, −4.065708950190296, −3.398111237879318, −2.470238362517657, −2.111483879994781, −1.498070697826239, −0.7744694638098056, 0,
0.7744694638098056, 1.498070697826239, 2.111483879994781, 2.470238362517657, 3.398111237879318, 4.065708950190296, 4.490886974829638, 5.174352323363938, 5.941350783971278, 6.233967830205165, 6.571955517078663, 7.027072003377404, 7.649108491797431, 8.358067540696352, 8.934265028358095, 9.106974467899618, 9.721213247187618, 10.08308492450342, 10.74734079937815, 10.87702185701595, 11.60825323699230, 12.00841213637971, 12.41344512942106, 13.00014249451384, 13.52935188132935
