| L(s) = 1 | + 2-s − 3-s + 4-s + 5-s − 6-s − 2·7-s + 8-s + 9-s + 10-s − 3·11-s − 12-s − 4·13-s − 2·14-s − 15-s + 16-s − 3·17-s + 18-s + 19-s + 20-s + 2·21-s − 3·22-s − 6·23-s − 24-s + 25-s − 4·26-s − 27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 0.577·3-s + 1/2·4-s + 0.447·5-s − 0.408·6-s − 0.755·7-s + 0.353·8-s + 1/3·9-s + 0.316·10-s − 0.904·11-s − 0.288·12-s − 1.10·13-s − 0.534·14-s − 0.258·15-s + 1/4·16-s − 0.727·17-s + 0.235·18-s + 0.229·19-s + 0.223·20-s + 0.436·21-s − 0.639·22-s − 1.25·23-s − 0.204·24-s + 1/5·25-s − 0.784·26-s − 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 55470 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 43 | \( 1 \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 + 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 3 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 9 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−14.98332403280032, −14.34960639755194, −13.69920356003369, −13.43009450193388, −12.75323582915808, −12.57758525295383, −11.91690651447910, −11.49432523954256, −10.74751288292874, −10.42459973910764, −9.783742587577754, −9.533515056904200, −8.701603442267652, −7.987406445188273, −7.362573636402387, −6.936366608412977, −6.375690210387134, −5.761391276520825, −5.365741530747513, −4.791767846569997, −4.249531619843229, −3.363289976595874, −2.964662998507557, −2.007170774292416, −1.716687058774999, 0, 0,
1.716687058774999, 2.007170774292416, 2.964662998507557, 3.363289976595874, 4.249531619843229, 4.791767846569997, 5.365741530747513, 5.761391276520825, 6.375690210387134, 6.936366608412977, 7.362573636402387, 7.987406445188273, 8.701603442267652, 9.533515056904200, 9.783742587577754, 10.42459973910764, 10.74751288292874, 11.49432523954256, 11.91690651447910, 12.57758525295383, 12.75323582915808, 13.43009450193388, 13.69920356003369, 14.34960639755194, 14.98332403280032
