| L(s) = 1 | + 3-s − 2·7-s + 9-s + 4·11-s − 2·13-s + 2·17-s + 4·19-s − 2·21-s + 2·23-s + 27-s + 10·29-s + 4·31-s + 4·33-s + 8·37-s − 2·39-s + 6·41-s + 43-s + 2·47-s − 3·49-s + 2·51-s + 12·53-s + 4·57-s − 4·59-s − 8·61-s − 2·63-s + 4·67-s + 2·69-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.755·7-s + 1/3·9-s + 1.20·11-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 0.436·21-s + 0.417·23-s + 0.192·27-s + 1.85·29-s + 0.718·31-s + 0.696·33-s + 1.31·37-s − 0.320·39-s + 0.937·41-s + 0.152·43-s + 0.291·47-s − 3/7·49-s + 0.280·51-s + 1.64·53-s + 0.529·57-s − 0.520·59-s − 1.02·61-s − 0.251·63-s + 0.488·67-s + 0.240·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 51600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.860952363\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.860952363\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 8 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.40759459587566, −14.02440436193388, −13.55815188341475, −13.01773547028685, −12.32818991095906, −12.05039146117819, −11.58711185439465, −10.79182373364102, −10.20936462231194, −9.701385923847409, −9.341656670186271, −8.886970858467201, −8.168602603850165, −7.670971610005652, −7.062660010089942, −6.543917802782275, −6.084411428290958, −5.346301935989330, −4.531020677527680, −4.169432540491237, −3.301041897332629, −2.936174327879450, −2.287186559928527, −1.203155564991172, −0.7659675056567394,
0.7659675056567394, 1.203155564991172, 2.287186559928527, 2.936174327879450, 3.301041897332629, 4.169432540491237, 4.531020677527680, 5.346301935989330, 6.084411428290958, 6.543917802782275, 7.062660010089942, 7.670971610005652, 8.168602603850165, 8.886970858467201, 9.341656670186271, 9.701385923847409, 10.20936462231194, 10.79182373364102, 11.58711185439465, 12.05039146117819, 12.32818991095906, 13.01773547028685, 13.55815188341475, 14.02440436193388, 14.40759459587566
