| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
| L(s) = 1 | + 2-s + 3-s + 4-s − 5-s + 6-s + 7-s + 8-s + 9-s − 10-s + 11-s + 12-s + 13-s + 14-s − 15-s + 16-s + 17-s + 18-s + 19-s − 20-s + 21-s + 22-s − 23-s + 24-s + 25-s + 26-s + 27-s + 28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1823 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1823 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(9.101029018\) |
| \(L(\frac12)\) |
\(\approx\) |
\(9.101029018\) |
| \(L(1)\) |
\(\approx\) |
\(3.311075258\) |
| \(L(1)\) |
\(\approx\) |
\(3.311075258\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 1823 | \( 1 \) |
| good | 2 | \( 1 + T \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 13 | \( 1 + T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 - T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 - T \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 - T \) |
| 61 | \( 1 - T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 - T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 - T \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−20.16967196069699599395664582533, −19.60281756257472408571484363007, −18.67960438119410745012595385623, −18.00420565790509111185150033498, −16.55476137490777932407498812085, −16.17421607807026189580652268853, −15.211213111348307647347394115251, −14.79268991419006996585974886329, −14.04554613900521084577691604587, −13.64277896963407973477717669819, −12.45383998488501860795769996888, −11.91045247324178930359563411739, −11.27727924759304218294425160824, −10.4000045608089174552794498227, −9.33445748326346008074217349728, −8.20650290422277392280989225175, −7.92800278609353841421073777834, −7.0765171635095906312956595510, −6.17265364770651558752973416627, −5.002079914196186459626688338596, −4.28286351869990426501949706550, −3.566445689833144810015624068873, −3.06987025596214717759750916692, −1.66571662857626764805436351986, −1.18841136167251337246709597811,
1.18841136167251337246709597811, 1.66571662857626764805436351986, 3.06987025596214717759750916692, 3.566445689833144810015624068873, 4.28286351869990426501949706550, 5.002079914196186459626688338596, 6.17265364770651558752973416627, 7.0765171635095906312956595510, 7.92800278609353841421073777834, 8.20650290422277392280989225175, 9.33445748326346008074217349728, 10.4000045608089174552794498227, 11.27727924759304218294425160824, 11.91045247324178930359563411739, 12.45383998488501860795769996888, 13.64277896963407973477717669819, 14.04554613900521084577691604587, 14.79268991419006996585974886329, 15.211213111348307647347394115251, 16.17421607807026189580652268853, 16.55476137490777932407498812085, 18.00420565790509111185150033498, 18.67960438119410745012595385623, 19.60281756257472408571484363007, 20.16967196069699599395664582533
