L(s) = 1 | − 10.9·2-s − 9·3-s + 87.0·4-s + 98.1·6-s − 180.·7-s − 600.·8-s + 81·9-s + 121·11-s − 783.·12-s − 61.0·13-s + 1.97e3·14-s + 3.76e3·16-s + 133.·17-s − 883.·18-s + 1.99e3·19-s + 1.62e3·21-s − 1.32e3·22-s + 445.·23-s + 5.40e3·24-s + 666.·26-s − 729·27-s − 1.57e4·28-s − 6.71e3·29-s − 8.73e3·31-s − 2.18e4·32-s − 1.08e3·33-s − 1.45e3·34-s + ⋯ |
L(s) = 1 | − 1.92·2-s − 0.577·3-s + 2.71·4-s + 1.11·6-s − 1.39·7-s − 3.31·8-s + 0.333·9-s + 0.301·11-s − 1.56·12-s − 0.100·13-s + 2.68·14-s + 3.67·16-s + 0.111·17-s − 0.642·18-s + 1.26·19-s + 0.804·21-s − 0.581·22-s + 0.175·23-s + 1.91·24-s + 0.193·26-s − 0.192·27-s − 3.78·28-s − 1.48·29-s − 1.63·31-s − 3.77·32-s − 0.174·33-s − 0.215·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 825 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.3546793447\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3546793447\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + 9T \) |
| 5 | \( 1 \) |
| 11 | \( 1 - 121T \) |
good | 2 | \( 1 + 10.9T + 32T^{2} \) |
| 7 | \( 1 + 180.T + 1.68e4T^{2} \) |
| 13 | \( 1 + 61.0T + 3.71e5T^{2} \) |
| 17 | \( 1 - 133.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.99e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 445.T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.71e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 8.73e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.49e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.98e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 6.57e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 1.91e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 2.00e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.06e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 8.34e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.41e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 4.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.79e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 2.13e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 9.30e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.30e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.68e5T + 8.58e9T^{2} \) |
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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
−9.434312417697342521076101141520, −9.040675710715031379748582113692, −7.60239947188951337672814212810, −7.25453050776198457455515806197, −6.25038527125330992484175365294, −5.65254537854282210002724126772, −3.65015341551642388709726777097, −2.63940337959941905672016699196, −1.36491137688305928849167539966, −0.39814640627144420110330139099,
0.39814640627144420110330139099, 1.36491137688305928849167539966, 2.63940337959941905672016699196, 3.65015341551642388709726777097, 5.65254537854282210002724126772, 6.25038527125330992484175365294, 7.25453050776198457455515806197, 7.60239947188951337672814212810, 9.040675710715031379748582113692, 9.434312417697342521076101141520