| L(s) = 1 | + 1.24·2-s − 1.80·3-s + 0.554·4-s − 2.24·6-s + 7-s − 0.554·8-s + 2.24·9-s − 0.445·11-s − 0.999·12-s + 1.24·13-s + 1.24·14-s − 1.24·16-s + 1.24·17-s + 2.80·18-s − 1.80·19-s − 1.80·21-s − 0.554·22-s − 1.80·23-s + 1.00·24-s + 25-s + 1.55·26-s − 2.24·27-s + 0.554·28-s + 2·31-s − 0.999·32-s + 0.801·33-s + 1.55·34-s + ⋯ |
| L(s) = 1 | + 1.24·2-s − 1.80·3-s + 0.554·4-s − 2.24·6-s + 7-s − 0.554·8-s + 2.24·9-s − 0.445·11-s − 0.999·12-s + 1.24·13-s + 1.24·14-s − 1.24·16-s + 1.24·17-s + 2.80·18-s − 1.80·19-s − 1.80·21-s − 0.554·22-s − 1.80·23-s + 1.00·24-s + 25-s + 1.55·26-s − 2.24·27-s + 0.554·28-s + 2·31-s − 0.999·32-s + 0.801·33-s + 1.55·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3143 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{1}{2})\) |
\(\approx\) |
\(1.378109468\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.378109468\) |
| \(L(1)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 - T \) |
| 449 | \( 1 - T \) |
| good | 2 | \( 1 - 1.24T + T^{2} \) |
| 3 | \( 1 + 1.80T + T^{2} \) |
| 5 | \( 1 - T^{2} \) |
| 11 | \( 1 + 0.445T + T^{2} \) |
| 13 | \( 1 - 1.24T + T^{2} \) |
| 17 | \( 1 - 1.24T + T^{2} \) |
| 19 | \( 1 + 1.80T + T^{2} \) |
| 23 | \( 1 + 1.80T + T^{2} \) |
| 29 | \( 1 - T^{2} \) |
| 31 | \( 1 - 2T + T^{2} \) |
| 37 | \( 1 - T^{2} \) |
| 41 | \( 1 - T^{2} \) |
| 43 | \( 1 - T^{2} \) |
| 47 | \( 1 - 2T + T^{2} \) |
| 53 | \( 1 - 1.24T + T^{2} \) |
| 59 | \( 1 - T^{2} \) |
| 61 | \( 1 - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 - T^{2} \) |
| 73 | \( 1 + 0.445T + T^{2} \) |
| 79 | \( 1 - T^{2} \) |
| 83 | \( 1 + 0.445T + T^{2} \) |
| 89 | \( 1 - T^{2} \) |
| 97 | \( 1 - T^{2} \) |
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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
−8.660306940193218393149331126458, −8.044281281253034019983132289549, −6.90471674365268834841857992503, −6.14425273475762130654412483662, −5.78692097719207461467519664022, −5.09195056784068736394861319883, −4.33452882314584084982079255891, −3.90369303236263724965681787063, −2.34017134277361786565248690116, −0.996849997227304474296876030169,
0.996849997227304474296876030169, 2.34017134277361786565248690116, 3.90369303236263724965681787063, 4.33452882314584084982079255891, 5.09195056784068736394861319883, 5.78692097719207461467519664022, 6.14425273475762130654412483662, 6.90471674365268834841857992503, 8.044281281253034019983132289549, 8.660306940193218393149331126458
