| L(s) = 1 | − 3-s − 4·7-s + 9-s − 2·11-s + 4·21-s − 8·23-s − 27-s − 2·29-s + 2·31-s + 2·33-s − 8·37-s − 2·41-s − 4·43-s + 9·49-s − 6·53-s − 14·59-s + 14·61-s − 4·63-s + 4·67-s + 8·69-s − 8·71-s + 10·73-s + 8·77-s − 6·79-s + 81-s + 12·83-s + 2·87-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s − 0.603·11-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 0.371·29-s + 0.359·31-s + 0.348·33-s − 1.31·37-s − 0.312·41-s − 0.609·43-s + 9/7·49-s − 0.824·53-s − 1.82·59-s + 1.79·61-s − 0.503·63-s + 0.488·67-s + 0.963·69-s − 0.949·71-s + 1.17·73-s + 0.911·77-s − 0.675·79-s + 1/9·81-s + 1.31·83-s + 0.214·87-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4307813867\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4307813867\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 - 14 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−7.67944047954991206161328635002, −6.69823678335968126368532253480, −6.47906084915385450201777091873, −5.66313036326026227385633844381, −5.11085009618435412385876268278, −4.09538191180020215795540483755, −3.49151565720533217184322887878, −2.67691874595018125648470883032, −1.70693338555725248228296519128, −0.30976501829187384570699942362,
0.30976501829187384570699942362, 1.70693338555725248228296519128, 2.67691874595018125648470883032, 3.49151565720533217184322887878, 4.09538191180020215795540483755, 5.11085009618435412385876268278, 5.66313036326026227385633844381, 6.47906084915385450201777091873, 6.69823678335968126368532253480, 7.67944047954991206161328635002
