| L(s) = 1 | − 2-s + 4-s + 5-s − 0.0586·7-s − 8-s − 10-s + 1.24·11-s + 3.77·13-s + 0.0586·14-s + 16-s + 7.49·17-s + 3.77·19-s + 20-s − 1.24·22-s − 3.66·23-s + 25-s − 3.77·26-s − 0.0586·28-s − 0.280·29-s + 4.41·31-s − 32-s − 7.49·34-s − 0.0586·35-s − 37-s − 3.77·38-s − 40-s − 7.14·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447·5-s − 0.0221·7-s − 0.353·8-s − 0.316·10-s + 0.376·11-s + 1.04·13-s + 0.0156·14-s + 0.250·16-s + 1.81·17-s + 0.866·19-s + 0.223·20-s − 0.266·22-s − 0.763·23-s + 0.200·25-s − 0.741·26-s − 0.0110·28-s − 0.0520·29-s + 0.792·31-s − 0.176·32-s − 1.28·34-s − 0.00991·35-s − 0.164·37-s − 0.612·38-s − 0.158·40-s − 1.11·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3330 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.740984319\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.740984319\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 5 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| good | 7 | \( 1 + 0.0586T + 7T^{2} \) |
| 11 | \( 1 - 1.24T + 11T^{2} \) |
| 13 | \( 1 - 3.77T + 13T^{2} \) |
| 17 | \( 1 - 7.49T + 17T^{2} \) |
| 19 | \( 1 - 3.77T + 19T^{2} \) |
| 23 | \( 1 + 3.66T + 23T^{2} \) |
| 29 | \( 1 + 0.280T + 29T^{2} \) |
| 31 | \( 1 - 4.41T + 31T^{2} \) |
| 41 | \( 1 + 7.14T + 41T^{2} \) |
| 43 | \( 1 - 11.0T + 43T^{2} \) |
| 47 | \( 1 + 6.05T + 47T^{2} \) |
| 53 | \( 1 + 5.36T + 53T^{2} \) |
| 59 | \( 1 - 0.615T + 59T^{2} \) |
| 61 | \( 1 - 5.02T + 61T^{2} \) |
| 67 | \( 1 - 1.38T + 67T^{2} \) |
| 71 | \( 1 + 6.61T + 71T^{2} \) |
| 73 | \( 1 + 14.3T + 73T^{2} \) |
| 79 | \( 1 - 5.50T + 79T^{2} \) |
| 83 | \( 1 - 14.3T + 83T^{2} \) |
| 89 | \( 1 + 11.3T + 89T^{2} \) |
| 97 | \( 1 - 9.14T + 97T^{2} \) |
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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
−8.620460498992659946659114495242, −7.953589790173832916144583321870, −7.31777505102384733351484263063, −6.29089548225614149844878370250, −5.87305950764625975050599637789, −4.94241890599754201763308665923, −3.68059835768488342719873586267, −3.03763099521543882408875341693, −1.72949468085714150997656627203, −0.949106662803851428706954267434,
0.949106662803851428706954267434, 1.72949468085714150997656627203, 3.03763099521543882408875341693, 3.68059835768488342719873586267, 4.94241890599754201763308665923, 5.87305950764625975050599637789, 6.29089548225614149844878370250, 7.31777505102384733351484263063, 7.953589790173832916144583321870, 8.620460498992659946659114495242
