| L(s) = 1 | − 5.07·2-s + 8.66·3-s + 17.7·4-s − 43.9·6-s + 15.1·7-s − 49.3·8-s + 48.1·9-s + 12.6·11-s + 153.·12-s − 46.9·13-s − 76.6·14-s + 108.·16-s + 28.9·17-s − 244.·18-s − 19·19-s + 130.·21-s − 63.9·22-s + 112.·23-s − 427.·24-s + 238.·26-s + 183.·27-s + 267.·28-s + 295.·29-s − 57.6·31-s − 154.·32-s + 109.·33-s − 146.·34-s + ⋯ |
| L(s) = 1 | − 1.79·2-s + 1.66·3-s + 2.21·4-s − 2.99·6-s + 0.815·7-s − 2.17·8-s + 1.78·9-s + 0.345·11-s + 3.69·12-s − 1.00·13-s − 1.46·14-s + 1.69·16-s + 0.413·17-s − 3.19·18-s − 0.229·19-s + 1.36·21-s − 0.620·22-s + 1.01·23-s − 3.63·24-s + 1.79·26-s + 1.30·27-s + 1.80·28-s + 1.88·29-s − 0.334·31-s − 0.855·32-s + 0.577·33-s − 0.741·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.792999954\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.792999954\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 19 | \( 1 + 19T \) |
| good | 2 | \( 1 + 5.07T + 8T^{2} \) |
| 3 | \( 1 - 8.66T + 27T^{2} \) |
| 7 | \( 1 - 15.1T + 343T^{2} \) |
| 11 | \( 1 - 12.6T + 1.33e3T^{2} \) |
| 13 | \( 1 + 46.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 28.9T + 4.91e3T^{2} \) |
| 23 | \( 1 - 112.T + 1.21e4T^{2} \) |
| 29 | \( 1 - 295.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 57.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 341.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 327.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 139.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 296.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 459.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 232.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 320.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 9.54T + 3.57e5T^{2} \) |
| 73 | \( 1 + 320.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 89.2T + 4.93e5T^{2} \) |
| 83 | \( 1 - 439.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 883.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 1.70e3T + 9.12e5T^{2} \) |
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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
−10.10798712722594660623811537153, −9.505831241616103377866340785823, −8.762422921597023549809201507134, −8.074038146281281106556226935227, −7.54716449145074726078191911488, −6.61675634312065184113569977842, −4.66034568682703804988250649926, −3.03238743106983099750173163075, −2.17075433587517844871797033346, −1.08247765685109220844021774056,
1.08247765685109220844021774056, 2.17075433587517844871797033346, 3.03238743106983099750173163075, 4.66034568682703804988250649926, 6.61675634312065184113569977842, 7.54716449145074726078191911488, 8.074038146281281106556226935227, 8.762422921597023549809201507134, 9.505831241616103377866340785823, 10.10798712722594660623811537153
