| L(s) = 1 | + 7.73·2-s − 12.3·3-s + 27.7·4-s − 25·5-s − 95.7·6-s + 224.·7-s − 32.6·8-s − 89.6·9-s − 193.·10-s + 121·11-s − 344.·12-s + 167.·13-s + 1.73e3·14-s + 309.·15-s − 1.14e3·16-s − 777.·17-s − 692.·18-s − 361·19-s − 694.·20-s − 2.78e3·21-s + 935.·22-s + 1.90e3·23-s + 403.·24-s + 625·25-s + 1.29e3·26-s + 4.11e3·27-s + 6.23e3·28-s + ⋯ |
| L(s) = 1 | + 1.36·2-s − 0.794·3-s + 0.868·4-s − 0.447·5-s − 1.08·6-s + 1.73·7-s − 0.180·8-s − 0.368·9-s − 0.611·10-s + 0.301·11-s − 0.689·12-s + 0.274·13-s + 2.36·14-s + 0.355·15-s − 1.11·16-s − 0.652·17-s − 0.504·18-s − 0.229·19-s − 0.388·20-s − 1.37·21-s + 0.412·22-s + 0.750·23-s + 0.143·24-s + 0.200·25-s + 0.375·26-s + 1.08·27-s + 1.50·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1045 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.474076100\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.474076100\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 + 25T \) |
| 11 | \( 1 - 121T \) |
| 19 | \( 1 + 361T \) |
| good | 2 | \( 1 - 7.73T + 32T^{2} \) |
| 3 | \( 1 + 12.3T + 243T^{2} \) |
| 7 | \( 1 - 224.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 167.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 777.T + 1.41e6T^{2} \) |
| 23 | \( 1 - 1.90e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 2.19e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.84e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.92e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 3.20e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 8.47e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.18e4T + 4.18e8T^{2} \) |
| 59 | \( 1 - 7.05e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.88e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.72e3T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 8.27e3T + 2.07e9T^{2} \) |
| 79 | \( 1 + 7.86e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 7.79e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 9.98e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 9.95e4T + 8.58e9T^{2} \) |
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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
−8.911381982965856093180414608447, −8.376487203990962888812543014030, −7.24582458420631636014094178950, −6.33581395827360189125476643397, −5.52227634623963515554117586659, −4.77936913132636778158797610297, −4.33962510409178512618957212077, −3.15799414982144315028540600327, −1.96487612086898768001709161173, −0.68113868251437454606821131908,
0.68113868251437454606821131908, 1.96487612086898768001709161173, 3.15799414982144315028540600327, 4.33962510409178512618957212077, 4.77936913132636778158797610297, 5.52227634623963515554117586659, 6.33581395827360189125476643397, 7.24582458420631636014094178950, 8.376487203990962888812543014030, 8.911381982965856093180414608447
