| L(s) = 1 | − 2-s + 4-s + 2·5-s + 7-s − 8-s − 2·10-s − 14-s + 16-s + 17-s + 2·20-s − 4·23-s − 25-s + 28-s − 2·31-s − 32-s − 34-s + 2·35-s + 4·37-s − 2·40-s − 6·41-s + 12·43-s + 4·46-s + 6·47-s + 49-s + 50-s − 8·53-s − 56-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.377·7-s − 0.353·8-s − 0.632·10-s − 0.267·14-s + 1/4·16-s + 0.242·17-s + 0.447·20-s − 0.834·23-s − 1/5·25-s + 0.188·28-s − 0.359·31-s − 0.176·32-s − 0.171·34-s + 0.338·35-s + 0.657·37-s − 0.316·40-s − 0.937·41-s + 1.82·43-s + 0.589·46-s + 0.875·47-s + 1/7·49-s + 0.141·50-s − 1.09·53-s − 0.133·56-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 259182 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 17 | \( 1 - T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 4 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 - 6 T + p T^{2} \) |
| 53 | \( 1 + 8 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 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
−12.88447253247197, −12.63287852701240, −12.10684004081290, −11.48369475657700, −11.22481315724449, −10.60935378684418, −10.24649142313747, −9.724440630670101, −9.443277285416691, −8.941142257496235, −8.391681972399289, −7.938027384082732, −7.524762724612534, −6.986980579313514, −6.359140212358129, −6.027913447825311, −5.440309579928249, −5.140061736747305, −4.184979454931135, −3.944767323251261, −3.037866817194069, −2.515950331822714, −1.996889658194994, −1.485315660062099, −0.8499024576327194, 0,
0.8499024576327194, 1.485315660062099, 1.996889658194994, 2.515950331822714, 3.037866817194069, 3.944767323251261, 4.184979454931135, 5.140061736747305, 5.440309579928249, 6.027913447825311, 6.359140212358129, 6.986980579313514, 7.524762724612534, 7.938027384082732, 8.391681972399289, 8.941142257496235, 9.443277285416691, 9.724440630670101, 10.24649142313747, 10.60935378684418, 11.22481315724449, 11.48369475657700, 12.10684004081290, 12.63287852701240, 12.88447253247197
