| L(s) = 1 | − 2-s + 4-s + 1.28·5-s − 0.550·7-s − 8-s − 1.28·10-s − 5.68·11-s − 3.95·13-s + 0.550·14-s + 16-s − 0.585·17-s + 5.61·19-s + 1.28·20-s + 5.68·22-s − 0.934·23-s − 3.34·25-s + 3.95·26-s − 0.550·28-s − 5.91·29-s − 5.94·31-s − 32-s + 0.585·34-s − 0.706·35-s + 3.47·37-s − 5.61·38-s − 1.28·40-s + 10.2·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.574·5-s − 0.207·7-s − 0.353·8-s − 0.406·10-s − 1.71·11-s − 1.09·13-s + 0.147·14-s + 0.250·16-s − 0.142·17-s + 1.28·19-s + 0.287·20-s + 1.21·22-s − 0.194·23-s − 0.669·25-s + 0.775·26-s − 0.103·28-s − 1.09·29-s − 1.06·31-s − 0.176·32-s + 0.100·34-s − 0.119·35-s + 0.571·37-s − 0.911·38-s − 0.203·40-s + 1.60·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4842 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.9660460743\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.9660460743\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 269 | \( 1 - T \) |
| good | 5 | \( 1 - 1.28T + 5T^{2} \) |
| 7 | \( 1 + 0.550T + 7T^{2} \) |
| 11 | \( 1 + 5.68T + 11T^{2} \) |
| 13 | \( 1 + 3.95T + 13T^{2} \) |
| 17 | \( 1 + 0.585T + 17T^{2} \) |
| 19 | \( 1 - 5.61T + 19T^{2} \) |
| 23 | \( 1 + 0.934T + 23T^{2} \) |
| 29 | \( 1 + 5.91T + 29T^{2} \) |
| 31 | \( 1 + 5.94T + 31T^{2} \) |
| 37 | \( 1 - 3.47T + 37T^{2} \) |
| 41 | \( 1 - 10.2T + 41T^{2} \) |
| 43 | \( 1 - 12.2T + 43T^{2} \) |
| 47 | \( 1 + 5.18T + 47T^{2} \) |
| 53 | \( 1 + 0.451T + 53T^{2} \) |
| 59 | \( 1 - 9.17T + 59T^{2} \) |
| 61 | \( 1 - 9.23T + 61T^{2} \) |
| 67 | \( 1 - 7.02T + 67T^{2} \) |
| 71 | \( 1 + 5.44T + 71T^{2} \) |
| 73 | \( 1 + 3.07T + 73T^{2} \) |
| 79 | \( 1 - 9.84T + 79T^{2} \) |
| 83 | \( 1 + 8.36T + 83T^{2} \) |
| 89 | \( 1 - 10.9T + 89T^{2} \) |
| 97 | \( 1 - 8.15T + 97T^{2} \) |
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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
−8.144403116795018406331829432678, −7.48826677125577948056681412287, −7.24813858478891482456163023339, −5.93503825424259215735031631143, −5.56211749563877075084893876197, −4.76617818950983869225959525703, −3.52449313228998430977728777940, −2.54695041329718493544931333273, −2.05838815689128174348208464769, −0.56935703370778779945007646593,
0.56935703370778779945007646593, 2.05838815689128174348208464769, 2.54695041329718493544931333273, 3.52449313228998430977728777940, 4.76617818950983869225959525703, 5.56211749563877075084893876197, 5.93503825424259215735031631143, 7.24813858478891482456163023339, 7.48826677125577948056681412287, 8.144403116795018406331829432678
