| L(s) = 1 | − 2-s − 4-s + 3·5-s + 3·8-s − 3·10-s + 5·11-s + 6·13-s − 16-s − 6·17-s + 3·19-s − 3·20-s − 5·22-s + 23-s + 4·25-s − 6·26-s − 2·29-s − 3·31-s − 5·32-s + 6·34-s + 3·37-s − 3·38-s + 9·40-s − 9·41-s + 6·43-s − 5·44-s − 46-s + 6·47-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1/2·4-s + 1.34·5-s + 1.06·8-s − 0.948·10-s + 1.50·11-s + 1.66·13-s − 1/4·16-s − 1.45·17-s + 0.688·19-s − 0.670·20-s − 1.06·22-s + 0.208·23-s + 4/5·25-s − 1.17·26-s − 0.371·29-s − 0.538·31-s − 0.883·32-s + 1.02·34-s + 0.493·37-s − 0.486·38-s + 1.42·40-s − 1.40·41-s + 0.914·43-s − 0.753·44-s − 0.147·46-s + 0.875·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.491293298\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.491293298\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + T + p T^{2} \) |
| 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 - 3 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 12 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 97 | \( 1 - 12 T + p T^{2} \) |
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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
−9.309598966834482564926917981283, −9.077294699564776327557372455758, −8.413589450998261439447207221768, −7.13167564560012152294416600100, −6.34215171511087571525667957813, −5.63752173484997695290532171122, −4.47191624657260288271626711721, −3.59632548449757607506915649618, −1.92610841189085090066739675799, −1.12292208190814267208777122415,
1.12292208190814267208777122415, 1.92610841189085090066739675799, 3.59632548449757607506915649618, 4.47191624657260288271626711721, 5.63752173484997695290532171122, 6.34215171511087571525667957813, 7.13167564560012152294416600100, 8.413589450998261439447207221768, 9.077294699564776327557372455758, 9.309598966834482564926917981283
