| L(s) = 1 | + 3-s + 2·5-s + 9-s + 4·11-s + 2·15-s + 2·17-s − 4·19-s + 8·23-s − 25-s + 27-s + 6·29-s + 8·31-s + 4·33-s − 6·37-s + 6·41-s − 4·43-s + 2·45-s − 7·49-s + 2·51-s − 2·53-s + 8·55-s − 4·57-s + 4·59-s − 2·61-s − 4·67-s + 8·69-s + 8·71-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.516·15-s + 0.485·17-s − 0.917·19-s + 1.66·23-s − 1/5·25-s + 0.192·27-s + 1.11·29-s + 1.43·31-s + 0.696·33-s − 0.986·37-s + 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s + 0.280·51-s − 0.274·53-s + 1.07·55-s − 0.529·57-s + 0.520·59-s − 0.256·61-s − 0.488·67-s + 0.963·69-s + 0.949·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.740344210\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.740344210\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 8 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.960162028579208482087007383582, −6.91770145820571465693225446041, −6.57343459837836202154520763791, −5.86703627689836153350861368375, −4.93339238468587225729143333287, −4.31433086854583759968075986340, −3.37399467735269751206047637641, −2.68692328900224142078971873167, −1.75453231670645219793496526812, −1.00382685160646478859467959828,
1.00382685160646478859467959828, 1.75453231670645219793496526812, 2.68692328900224142078971873167, 3.37399467735269751206047637641, 4.31433086854583759968075986340, 4.93339238468587225729143333287, 5.86703627689836153350861368375, 6.57343459837836202154520763791, 6.91770145820571465693225446041, 7.960162028579208482087007383582
