| L(s) = 1 | + 2-s + 4-s + 2·5-s + 8-s + 2·10-s + 6·11-s + 6·13-s + 16-s − 2·17-s − 4·19-s + 2·20-s + 6·22-s − 23-s − 25-s + 6·26-s − 2·29-s − 6·31-s + 32-s − 2·34-s + 8·37-s − 4·38-s + 2·40-s + 2·43-s + 6·44-s − 46-s − 6·47-s − 50-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 0.894·5-s + 0.353·8-s + 0.632·10-s + 1.80·11-s + 1.66·13-s + 1/4·16-s − 0.485·17-s − 0.917·19-s + 0.447·20-s + 1.27·22-s − 0.208·23-s − 1/5·25-s + 1.17·26-s − 0.371·29-s − 1.07·31-s + 0.176·32-s − 0.342·34-s + 1.31·37-s − 0.648·38-s + 0.316·40-s + 0.304·43-s + 0.904·44-s − 0.147·46-s − 0.875·47-s − 0.141·50-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20286 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(5.700481174\) |
| \(L(\frac12)\) |
\(\approx\) |
\(5.700481174\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 7 | \( 1 \) |
| 23 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.59943313174612, −14.80554690140506, −14.60656526195606, −13.93189266034989, −13.54405035874560, −12.93067394784110, −12.64133934572079, −11.59368902335122, −11.42458578998074, −10.83052878901854, −10.16126490122501, −9.364780568971978, −9.076312622863317, −8.407459101325716, −7.670820744598356, −6.659073028000701, −6.412643860543325, −6.000223653562042, −5.309315867518304, −4.370590727829853, −3.867688395044688, −3.405944126566463, −2.197290875594444, −1.752437098540800, −0.9256640992084051,
0.9256640992084051, 1.752437098540800, 2.197290875594444, 3.405944126566463, 3.867688395044688, 4.370590727829853, 5.309315867518304, 6.000223653562042, 6.412643860543325, 6.659073028000701, 7.670820744598356, 8.407459101325716, 9.076312622863317, 9.364780568971978, 10.16126490122501, 10.83052878901854, 11.42458578998074, 11.59368902335122, 12.64133934572079, 12.93067394784110, 13.54405035874560, 13.93189266034989, 14.60656526195606, 14.80554690140506, 15.59943313174612
