| L(s) = 1 | − 1.37·2-s − 6.10·4-s + 7·7-s + 19.4·8-s − 15.2·11-s + 76.7·13-s − 9.63·14-s + 22.1·16-s + 96.7·17-s − 14.1·19-s + 21.0·22-s + 75.7·23-s − 105.·26-s − 42.7·28-s − 89.9·29-s + 289.·31-s − 185.·32-s − 133.·34-s + 14.2·37-s + 19.4·38-s − 318.·41-s + 389.·43-s + 93.3·44-s − 104.·46-s − 228.·47-s + 49·49-s − 468.·52-s + ⋯ |
| L(s) = 1 | − 0.486·2-s − 0.763·4-s + 0.377·7-s + 0.857·8-s − 0.418·11-s + 1.63·13-s − 0.183·14-s + 0.345·16-s + 1.37·17-s − 0.171·19-s + 0.203·22-s + 0.686·23-s − 0.797·26-s − 0.288·28-s − 0.576·29-s + 1.67·31-s − 1.02·32-s − 0.671·34-s + 0.0632·37-s + 0.0832·38-s − 1.21·41-s + 1.38·43-s + 0.319·44-s − 0.333·46-s − 0.710·47-s + 0.142·49-s − 1.25·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1575 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.646401973\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.646401973\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 - 7T \) |
| good | 2 | \( 1 + 1.37T + 8T^{2} \) |
| 11 | \( 1 + 15.2T + 1.33e3T^{2} \) |
| 13 | \( 1 - 76.7T + 2.19e3T^{2} \) |
| 17 | \( 1 - 96.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 14.1T + 6.85e3T^{2} \) |
| 23 | \( 1 - 75.7T + 1.21e4T^{2} \) |
| 29 | \( 1 + 89.9T + 2.43e4T^{2} \) |
| 31 | \( 1 - 289.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 14.2T + 5.06e4T^{2} \) |
| 41 | \( 1 + 318.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 389.T + 7.95e4T^{2} \) |
| 47 | \( 1 + 228.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 679.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 398.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 146.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 291.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 333.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 891.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 416.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 814.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 650.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.58e3T + 9.12e5T^{2} \) |
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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
−9.034798841273159948436936801765, −8.125120221016582664363376616248, −7.955645674393116545397028865576, −6.69071208284956395267225826893, −5.70068427247550463944700885442, −4.95646536809536067163428523803, −3.98506986205537041382411820736, −3.11579181911685273235131350913, −1.52414060992694602984362130928, −0.74725289111070947176102852662,
0.74725289111070947176102852662, 1.52414060992694602984362130928, 3.11579181911685273235131350913, 3.98506986205537041382411820736, 4.95646536809536067163428523803, 5.70068427247550463944700885442, 6.69071208284956395267225826893, 7.955645674393116545397028865576, 8.125120221016582664363376616248, 9.034798841273159948436936801765
