| L(s) = 1 | − 3-s − 2·7-s − 2·9-s + 3·11-s − 4·13-s − 3·17-s − 5·19-s + 2·21-s − 6·23-s + 5·27-s − 2·31-s − 3·33-s + 2·37-s + 4·39-s − 3·41-s + 4·43-s − 12·47-s − 3·49-s + 3·51-s + 6·53-s + 5·57-s + 2·61-s + 4·63-s + 13·67-s + 6·69-s − 12·71-s + 11·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.755·7-s − 2/3·9-s + 0.904·11-s − 1.10·13-s − 0.727·17-s − 1.14·19-s + 0.436·21-s − 1.25·23-s + 0.962·27-s − 0.359·31-s − 0.522·33-s + 0.328·37-s + 0.640·39-s − 0.468·41-s + 0.609·43-s − 1.75·47-s − 3/7·49-s + 0.420·51-s + 0.824·53-s + 0.662·57-s + 0.256·61-s + 0.503·63-s + 1.58·67-s + 0.722·69-s − 1.42·71-s + 1.28·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 9 T + p T^{2} \) |
| 89 | \( 1 - 15 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−10.89154339860195613441988561800, −9.919702445885858877180142258118, −9.107735829970857706936675027245, −8.082389583115066239879169681269, −6.71237324686107378754912317869, −6.21926731108516302667159437545, −4.98537291304699817306164733550, −3.80421461859967166401405341307, −2.32235507105178670692643054585, 0,
2.32235507105178670692643054585, 3.80421461859967166401405341307, 4.98537291304699817306164733550, 6.21926731108516302667159437545, 6.71237324686107378754912317869, 8.082389583115066239879169681269, 9.107735829970857706936675027245, 9.919702445885858877180142258118, 10.89154339860195613441988561800
