| L(s) = 1 | + 3-s − 7-s − 2·9-s + 11-s + 6·13-s − 7·17-s − 19-s − 21-s − 8·23-s − 5·27-s − 6·29-s − 4·31-s + 33-s + 8·37-s + 6·39-s − 5·41-s − 6·47-s + 49-s − 7·51-s + 4·53-s − 57-s + 4·59-s + 6·61-s + 2·63-s + 5·67-s − 8·69-s − 14·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.377·7-s − 2/3·9-s + 0.301·11-s + 1.66·13-s − 1.69·17-s − 0.229·19-s − 0.218·21-s − 1.66·23-s − 0.962·27-s − 1.11·29-s − 0.718·31-s + 0.174·33-s + 1.31·37-s + 0.960·39-s − 0.780·41-s − 0.875·47-s + 1/7·49-s − 0.980·51-s + 0.549·53-s − 0.132·57-s + 0.520·59-s + 0.768·61-s + 0.251·63-s + 0.610·67-s − 0.963·69-s − 1.66·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 11 | \( 1 - T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 7 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 5 T + p T^{2} \) |
| 43 | \( 1 + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 5 T + p T^{2} \) |
| 71 | \( 1 + 14 T + p T^{2} \) |
| 73 | \( 1 - 15 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 + T + p T^{2} \) |
| 89 | \( 1 + 3 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−8.491343516237986420780812002390, −7.915023309241502299863215263071, −6.77699400648325343050669087655, −6.18643352994375537394425759481, −5.50284117863848417639671756489, −4.08538620796960676469495063267, −3.74668446305306253097899775917, −2.59881990082411890234805813183, −1.71597135942776495338536943488, 0,
1.71597135942776495338536943488, 2.59881990082411890234805813183, 3.74668446305306253097899775917, 4.08538620796960676469495063267, 5.50284117863848417639671756489, 6.18643352994375537394425759481, 6.77699400648325343050669087655, 7.915023309241502299863215263071, 8.491343516237986420780812002390
