| L(s) = 1 | − 2-s − 3·3-s + 4-s + 3·6-s + 7-s − 8-s + 6·9-s + 5·11-s − 3·12-s − 14-s + 16-s − 17-s − 6·18-s + 3·19-s − 3·21-s − 5·22-s + 3·24-s − 9·27-s + 28-s − 6·29-s + 4·31-s − 32-s − 15·33-s + 34-s + 6·36-s − 8·37-s − 3·38-s + ⋯ |
| L(s) = 1 | − 0.707·2-s − 1.73·3-s + 1/2·4-s + 1.22·6-s + 0.377·7-s − 0.353·8-s + 2·9-s + 1.50·11-s − 0.866·12-s − 0.267·14-s + 1/4·16-s − 0.242·17-s − 1.41·18-s + 0.688·19-s − 0.654·21-s − 1.06·22-s + 0.612·24-s − 1.73·27-s + 0.188·28-s − 1.11·29-s + 0.718·31-s − 0.176·32-s − 2.61·33-s + 0.171·34-s + 36-s − 1.31·37-s − 0.486·38-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59150 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 17 | \( 1 + T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + 2 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 + 2 T + p T^{2} \) |
| 83 | \( 1 + 11 T + p T^{2} \) |
| 89 | \( 1 - 11 T + p T^{2} \) |
| 97 | \( 1 - 10 T + p T^{2} \) |
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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
−14.72058882599422, −14.04450441463771, −13.54441125805403, −12.86251794205886, −12.17422220632617, −11.91605355286900, −11.44964483980104, −11.23890578755205, −10.45196536831581, −10.11924807979368, −9.594056007573981, −8.926323861845629, −8.518778527923117, −7.632697206834083, −7.182200592366685, −6.502968179094763, −6.436427954104847, −5.583732095496035, −5.118753294430126, −4.575934234313697, −3.793209118508875, −3.250985733572658, −1.898589755475684, −1.539817869434662, −0.8135799718974287, 0,
0.8135799718974287, 1.539817869434662, 1.898589755475684, 3.250985733572658, 3.793209118508875, 4.575934234313697, 5.118753294430126, 5.583732095496035, 6.436427954104847, 6.502968179094763, 7.182200592366685, 7.632697206834083, 8.518778527923117, 8.926323861845629, 9.594056007573981, 10.11924807979368, 10.45196536831581, 11.23890578755205, 11.44964483980104, 11.91605355286900, 12.17422220632617, 12.86251794205886, 13.54441125805403, 14.04450441463771, 14.72058882599422
