| L(s) = 1 | − 5-s − 11-s + 2·13-s + 2·17-s + 6·19-s + 25-s − 2·29-s − 4·31-s − 2·37-s + 4·43-s + 8·47-s + 4·53-s + 55-s − 14·59-s − 14·61-s − 2·65-s − 2·67-s + 8·71-s + 10·73-s − 16·79-s + 4·83-s − 2·85-s + 10·89-s − 6·95-s − 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.301·11-s + 0.554·13-s + 0.485·17-s + 1.37·19-s + 1/5·25-s − 0.371·29-s − 0.718·31-s − 0.328·37-s + 0.609·43-s + 1.16·47-s + 0.549·53-s + 0.134·55-s − 1.82·59-s − 1.79·61-s − 0.248·65-s − 0.244·67-s + 0.949·71-s + 1.17·73-s − 1.80·79-s + 0.439·83-s − 0.216·85-s + 1.05·89-s − 0.615·95-s − 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 97020 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.093643729\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.093643729\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 4 T + p T^{2} \) |
| 59 | \( 1 + 14 T + p T^{2} \) |
| 61 | \( 1 + 14 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.76135262221837, −13.44500092518297, −12.68651796685972, −12.27910834884739, −11.93885279761544, −11.27509727000156, −10.84494253518651, −10.48675165771400, −9.767572430132700, −9.245952907365343, −8.946159696804478, −8.135758886099568, −7.775643623235478, −7.306990992845780, −6.834828085829528, −5.980978233432660, −5.662706462085667, −5.078373701749997, −4.429410275140154, −3.822395459085553, −3.259959771472439, −2.814910183174415, −1.892738774476849, −1.226551622149558, −0.4910528494430213,
0.4910528494430213, 1.226551622149558, 1.892738774476849, 2.814910183174415, 3.259959771472439, 3.822395459085553, 4.429410275140154, 5.078373701749997, 5.662706462085667, 5.980978233432660, 6.834828085829528, 7.306990992845780, 7.775643623235478, 8.135758886099568, 8.946159696804478, 9.245952907365343, 9.767572430132700, 10.48675165771400, 10.84494253518651, 11.27509727000156, 11.93885279761544, 12.27910834884739, 12.68651796685972, 13.44500092518297, 13.76135262221837
