| L(s) = 1 | − 0.732·2-s − 7.46·4-s − 16.9·7-s + 11.3·8-s + 11·11-s − 74.6·13-s + 12.3·14-s + 51.4·16-s − 82.7·17-s − 67.9·19-s − 8.05·22-s + 13.3·23-s + 54.6·26-s + 126.·28-s − 168.·29-s − 65.4·31-s − 128.·32-s + 60.6·34-s − 40.8·37-s + 49.7·38-s − 274.·41-s + 2.28·43-s − 82.1·44-s − 9.77·46-s + 71.8·47-s − 56.4·49-s + 557.·52-s + ⋯ |
| L(s) = 1 | − 0.258·2-s − 0.933·4-s − 0.914·7-s + 0.500·8-s + 0.301·11-s − 1.59·13-s + 0.236·14-s + 0.803·16-s − 1.18·17-s − 0.820·19-s − 0.0780·22-s + 0.121·23-s + 0.412·26-s + 0.852·28-s − 1.08·29-s − 0.379·31-s − 0.708·32-s + 0.305·34-s − 0.181·37-s + 0.212·38-s − 1.04·41-s + 0.00811·43-s − 0.281·44-s − 0.0313·46-s + 0.222·47-s − 0.164·49-s + 1.48·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2475 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.03710789437\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03710789437\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 0.732T + 8T^{2} \) |
| 7 | \( 1 + 16.9T + 343T^{2} \) |
| 13 | \( 1 + 74.6T + 2.19e3T^{2} \) |
| 17 | \( 1 + 82.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 67.9T + 6.85e3T^{2} \) |
| 23 | \( 1 - 13.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 168.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 65.4T + 2.97e4T^{2} \) |
| 37 | \( 1 + 40.8T + 5.06e4T^{2} \) |
| 41 | \( 1 + 274.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 2.28T + 7.95e4T^{2} \) |
| 47 | \( 1 - 71.8T + 1.03e5T^{2} \) |
| 53 | \( 1 + 149.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 545.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 101.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 411.T + 3.00e5T^{2} \) |
| 71 | \( 1 - 470.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 610.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 978.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 26.1T + 5.71e5T^{2} \) |
| 89 | \( 1 - 352.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 847.T + 9.12e5T^{2} \) |
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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
−8.845453733810344542760511320816, −7.84847147449465822319118412791, −7.08923631205891256304624426717, −6.36484366068527356967868924771, −5.34067623115730663760787883670, −4.57765227154553697146636426464, −3.85317271792736817146173714985, −2.80463247367345617896671556252, −1.71834942272120858071101274549, −0.085896833210309871970054469928,
0.085896833210309871970054469928, 1.71834942272120858071101274549, 2.80463247367345617896671556252, 3.85317271792736817146173714985, 4.57765227154553697146636426464, 5.34067623115730663760787883670, 6.36484366068527356967868924771, 7.08923631205891256304624426717, 7.84847147449465822319118412791, 8.845453733810344542760511320816
