| L(s) = 1 | − 1.65·2-s − 0.332·3-s − 5.25·4-s + 0.551·6-s + 21.9·8-s − 26.8·9-s + 69.5·11-s + 1.75·12-s + 68.4·13-s + 5.70·16-s + 104.·17-s + 44.5·18-s − 71.8·19-s − 115.·22-s + 101.·23-s − 7.30·24-s − 113.·26-s + 17.9·27-s − 114.·29-s + 73.6·31-s − 185.·32-s − 23.1·33-s − 172.·34-s + 141.·36-s + 200.·37-s + 119.·38-s − 22.7·39-s + ⋯ |
| L(s) = 1 | − 0.585·2-s − 0.0640·3-s − 0.657·4-s + 0.0375·6-s + 0.970·8-s − 0.995·9-s + 1.90·11-s + 0.0421·12-s + 1.45·13-s + 0.0890·16-s + 1.48·17-s + 0.583·18-s − 0.868·19-s − 1.11·22-s + 0.915·23-s − 0.0621·24-s − 0.854·26-s + 0.127·27-s − 0.734·29-s + 0.426·31-s − 1.02·32-s − 0.122·33-s − 0.871·34-s + 0.654·36-s + 0.892·37-s + 0.508·38-s − 0.0935·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1225 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.474934820\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.474934820\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 2 | \( 1 + 1.65T + 8T^{2} \) |
| 3 | \( 1 + 0.332T + 27T^{2} \) |
| 11 | \( 1 - 69.5T + 1.33e3T^{2} \) |
| 13 | \( 1 - 68.4T + 2.19e3T^{2} \) |
| 17 | \( 1 - 104.T + 4.91e3T^{2} \) |
| 19 | \( 1 + 71.8T + 6.85e3T^{2} \) |
| 23 | \( 1 - 101.T + 1.21e4T^{2} \) |
| 29 | \( 1 + 114.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 73.6T + 2.97e4T^{2} \) |
| 37 | \( 1 - 200.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 417.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 311.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 149.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 271.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 518.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 219.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 80.6T + 3.00e5T^{2} \) |
| 71 | \( 1 + 91.0T + 3.57e5T^{2} \) |
| 73 | \( 1 - 882.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 599.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 70.8T + 5.71e5T^{2} \) |
| 89 | \( 1 + 802.T + 7.04e5T^{2} \) |
| 97 | \( 1 - 145.T + 9.12e5T^{2} \) |
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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
−9.221130152890835921151695192579, −8.584901737368787215678314599034, −8.113507330754912603613701071938, −6.83629868741450525564396165015, −6.07930003907631308981923639333, −5.16541427449279344542748021451, −3.96235560569125488344845329640, −3.37765340622768250436856846294, −1.55220758283825098589228392622, −0.75892959219911703547652223131,
0.75892959219911703547652223131, 1.55220758283825098589228392622, 3.37765340622768250436856846294, 3.96235560569125488344845329640, 5.16541427449279344542748021451, 6.07930003907631308981923639333, 6.83629868741450525564396165015, 8.113507330754912603613701071938, 8.584901737368787215678314599034, 9.221130152890835921151695192579
