| L(s) = 1 | − 2-s + 4-s + 5-s − 8-s − 10-s − 5·11-s + 16-s − 4·17-s − 8·19-s + 20-s + 5·22-s + 4·23-s − 4·25-s + 5·29-s − 3·31-s − 32-s + 4·34-s − 4·37-s + 8·38-s − 40-s + 2·43-s − 5·44-s − 4·46-s − 6·47-s + 4·50-s + 9·53-s − 5·55-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.447·5-s − 0.353·8-s − 0.316·10-s − 1.50·11-s + 1/4·16-s − 0.970·17-s − 1.83·19-s + 0.223·20-s + 1.06·22-s + 0.834·23-s − 4/5·25-s + 0.928·29-s − 0.538·31-s − 0.176·32-s + 0.685·34-s − 0.657·37-s + 1.29·38-s − 0.158·40-s + 0.304·43-s − 0.753·44-s − 0.589·46-s − 0.875·47-s + 0.565·50-s + 1.23·53-s − 0.674·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 882 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 5 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 9 T + p T^{2} \) |
| 59 | \( 1 + 11 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 + 7 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 7 T + p T^{2} \) |
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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
−9.763477646741241139999396192878, −8.776310261118363154608709933551, −8.238671890386923809123178878617, −7.21435528320358379927581416164, −6.38468865530543948344255312711, −5.43475276494163016869526953658, −4.36865869014622242250082301422, −2.81104469657788282313974403144, −1.94506082349214662942482411882, 0,
1.94506082349214662942482411882, 2.81104469657788282313974403144, 4.36865869014622242250082301422, 5.43475276494163016869526953658, 6.38468865530543948344255312711, 7.21435528320358379927581416164, 8.238671890386923809123178878617, 8.776310261118363154608709933551, 9.763477646741241139999396192878
