| L(s) = 1 | − 2-s + 4-s − 5-s − 2·7-s − 8-s + 10-s + 11-s − 4·13-s + 2·14-s + 16-s − 17-s − 6·19-s − 20-s − 22-s + 2·23-s + 25-s + 4·26-s − 2·28-s + 6·29-s − 4·31-s − 32-s + 34-s + 2·35-s − 2·37-s + 6·38-s + 40-s + 6·41-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 1/2·4-s − 0.447·5-s − 0.755·7-s − 0.353·8-s + 0.316·10-s + 0.301·11-s − 1.10·13-s + 0.534·14-s + 1/4·16-s − 0.242·17-s − 1.37·19-s − 0.223·20-s − 0.213·22-s + 0.417·23-s + 1/5·25-s + 0.784·26-s − 0.377·28-s + 1.11·29-s − 0.718·31-s − 0.176·32-s + 0.171·34-s + 0.338·35-s − 0.328·37-s + 0.973·38-s + 0.158·40-s + 0.937·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16830 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−16.14940963265893, −15.73587694692979, −15.18571075730023, −14.61036505661765, −14.14939150200877, −13.27700145455612, −12.59913386197760, −12.36922151787253, −11.72922067864808, −10.93857836227078, −10.57174936327552, −9.963947232818078, −9.262721541814381, −8.907620356398440, −8.258537399151443, −7.429222063743404, −7.146386263185677, −6.365071034562125, −5.916800098580665, −4.863771799678897, −4.285633841193968, −3.474730280237911, −2.662543620740451, −2.089195921859799, −0.8554177841824049, 0,
0.8554177841824049, 2.089195921859799, 2.662543620740451, 3.474730280237911, 4.285633841193968, 4.863771799678897, 5.916800098580665, 6.365071034562125, 7.146386263185677, 7.429222063743404, 8.258537399151443, 8.907620356398440, 9.262721541814381, 9.963947232818078, 10.57174936327552, 10.93857836227078, 11.72922067864808, 12.36922151787253, 12.59913386197760, 13.27700145455612, 14.14939150200877, 14.61036505661765, 15.18571075730023, 15.73587694692979, 16.14940963265893
