| L(s) = 1 | − 1.53·2-s − 5.65·4-s − 8.69·5-s − 7·7-s + 20.9·8-s + 13.3·10-s + 11·11-s − 76.3·13-s + 10.7·14-s + 13.1·16-s − 39.7·17-s − 27.9·19-s + 49.1·20-s − 16.8·22-s − 87.2·23-s − 49.3·25-s + 117.·26-s + 39.5·28-s + 38.3·29-s − 186.·31-s − 187.·32-s + 60.8·34-s + 60.8·35-s − 218.·37-s + 42.8·38-s − 182.·40-s − 80.1·41-s + ⋯ |
| L(s) = 1 | − 0.541·2-s − 0.706·4-s − 0.778·5-s − 0.377·7-s + 0.924·8-s + 0.421·10-s + 0.301·11-s − 1.62·13-s + 0.204·14-s + 0.205·16-s − 0.566·17-s − 0.337·19-s + 0.549·20-s − 0.163·22-s − 0.790·23-s − 0.394·25-s + 0.882·26-s + 0.267·28-s + 0.245·29-s − 1.07·31-s − 1.03·32-s + 0.307·34-s + 0.294·35-s − 0.972·37-s + 0.183·38-s − 0.719·40-s − 0.305·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 693 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3527487740\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3527487740\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + 7T \) |
| 11 | \( 1 - 11T \) |
| good | 2 | \( 1 + 1.53T + 8T^{2} \) |
| 5 | \( 1 + 8.69T + 125T^{2} \) |
| 13 | \( 1 + 76.3T + 2.19e3T^{2} \) |
| 17 | \( 1 + 39.7T + 4.91e3T^{2} \) |
| 19 | \( 1 + 27.9T + 6.85e3T^{2} \) |
| 23 | \( 1 + 87.2T + 1.21e4T^{2} \) |
| 29 | \( 1 - 38.3T + 2.43e4T^{2} \) |
| 31 | \( 1 + 186.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 218.T + 5.06e4T^{2} \) |
| 41 | \( 1 + 80.1T + 6.89e4T^{2} \) |
| 43 | \( 1 + 35.1T + 7.95e4T^{2} \) |
| 47 | \( 1 - 282.T + 1.03e5T^{2} \) |
| 53 | \( 1 + 145.T + 1.48e5T^{2} \) |
| 59 | \( 1 + 91.0T + 2.05e5T^{2} \) |
| 61 | \( 1 - 808.T + 2.26e5T^{2} \) |
| 67 | \( 1 - 794.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 946.T + 3.57e5T^{2} \) |
| 73 | \( 1 - 801.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 890.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 559.T + 5.71e5T^{2} \) |
| 89 | \( 1 - 1.52e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 664.T + 9.12e5T^{2} \) |
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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
−9.913391569045315700428913678766, −9.243535522594333647318723097076, −8.376401220941887360481463832105, −7.58070792967589942241003918007, −6.83662884061121346242228080073, −5.41341969349810076460555909626, −4.44790086661074123933127047207, −3.64629289644576259721691776350, −2.09215771946091788090118919731, −0.35922587688180396532302366460,
0.35922587688180396532302366460, 2.09215771946091788090118919731, 3.64629289644576259721691776350, 4.44790086661074123933127047207, 5.41341969349810076460555909626, 6.83662884061121346242228080073, 7.58070792967589942241003918007, 8.376401220941887360481463832105, 9.243535522594333647318723097076, 9.913391569045315700428913678766
