| L(s) = 1 | + 2-s − 4-s − 2·5-s + 7-s − 3·8-s − 2·10-s + 14-s − 16-s + 2·17-s − 2·19-s + 2·20-s − 6·23-s − 25-s − 28-s + 29-s − 8·31-s + 5·32-s + 2·34-s − 2·35-s − 2·37-s − 2·38-s + 6·40-s + 6·41-s − 2·43-s − 6·46-s + 4·47-s + 49-s + ⋯ |
| L(s) = 1 | + 0.707·2-s − 1/2·4-s − 0.894·5-s + 0.377·7-s − 1.06·8-s − 0.632·10-s + 0.267·14-s − 1/4·16-s + 0.485·17-s − 0.458·19-s + 0.447·20-s − 1.25·23-s − 1/5·25-s − 0.188·28-s + 0.185·29-s − 1.43·31-s + 0.883·32-s + 0.342·34-s − 0.338·35-s − 0.328·37-s − 0.324·38-s + 0.948·40-s + 0.937·41-s − 0.304·43-s − 0.884·46-s + 0.583·47-s + 1/7·49-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 221067 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.498033647\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.498033647\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 \) |
| 29 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 8 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 - 10 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 4 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| 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
−12.98557201281870, −12.43055572022303, −12.15184731644882, −11.75849168437586, −11.14526876531540, −10.86225260423123, −10.10880642556753, −9.715575300716065, −9.176919160082507, −8.677306145185549, −8.123100522705120, −7.883064203944979, −7.326615674246064, −6.670561949880390, −6.111704489916586, −5.644239070899118, −5.093957100341908, −4.704699507595723, −4.001843105223707, −3.693199051170661, −3.440908715878732, −2.382581368673546, −2.060688882712044, −0.9934544828409617, −0.3528934216512016,
0.3528934216512016, 0.9934544828409617, 2.060688882712044, 2.382581368673546, 3.440908715878732, 3.693199051170661, 4.001843105223707, 4.704699507595723, 5.093957100341908, 5.644239070899118, 6.111704489916586, 6.670561949880390, 7.326615674246064, 7.883064203944979, 8.123100522705120, 8.677306145185549, 9.176919160082507, 9.715575300716065, 10.10880642556753, 10.86225260423123, 11.14526876531540, 11.75849168437586, 12.15184731644882, 12.43055572022303, 12.98557201281870
