| L(s) = 1 | − 3·3-s − 4·7-s + 6·9-s − 3·11-s − 6·13-s + 5·17-s + 19-s + 12·21-s − 23-s − 9·27-s − 8·29-s + 8·31-s + 9·33-s + 2·37-s + 18·39-s − 7·41-s − 4·43-s − 10·47-s + 9·49-s − 15·51-s − 12·53-s − 3·57-s − 4·59-s − 8·61-s − 24·63-s − 3·67-s + 3·69-s + ⋯ |
| L(s) = 1 | − 1.73·3-s − 1.51·7-s + 2·9-s − 0.904·11-s − 1.66·13-s + 1.21·17-s + 0.229·19-s + 2.61·21-s − 0.208·23-s − 1.73·27-s − 1.48·29-s + 1.43·31-s + 1.56·33-s + 0.328·37-s + 2.88·39-s − 1.09·41-s − 0.609·43-s − 1.45·47-s + 9/7·49-s − 2.10·51-s − 1.64·53-s − 0.397·57-s − 0.520·59-s − 1.02·61-s − 3.02·63-s − 0.366·67-s + 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 10 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 3 T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 6 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 14 T + p T^{2} \) |
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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
−6.92239377441840684816471747454, −6.24535621074205171772140621916, −5.71106077852309304519917165533, −5.07339716290305424227333036164, −4.55789642399542140102945496038, −3.39716192060488143464081627253, −2.75622102866174471385914253007, −1.45590458467864861829912255714, 0, 0,
1.45590458467864861829912255714, 2.75622102866174471385914253007, 3.39716192060488143464081627253, 4.55789642399542140102945496038, 5.07339716290305424227333036164, 5.71106077852309304519917165533, 6.24535621074205171772140621916, 6.92239377441840684816471747454
