| L(s) = 1 | − 0.948·2-s − 7.10·4-s + 5.36·5-s − 7·7-s + 14.3·8-s − 5.09·10-s − 11·11-s − 42.9·13-s + 6.64·14-s + 43.2·16-s + 60.8·17-s + 140.·19-s − 38.1·20-s + 10.4·22-s + 91.3·23-s − 96.1·25-s + 40.7·26-s + 49.7·28-s − 260.·29-s − 259.·31-s − 155.·32-s − 57.7·34-s − 37.5·35-s + 359.·37-s − 133.·38-s + 76.8·40-s + 320.·41-s + ⋯ |
| L(s) = 1 | − 0.335·2-s − 0.887·4-s + 0.480·5-s − 0.377·7-s + 0.633·8-s − 0.161·10-s − 0.301·11-s − 0.916·13-s + 0.126·14-s + 0.675·16-s + 0.868·17-s + 1.69·19-s − 0.426·20-s + 0.101·22-s + 0.828·23-s − 0.769·25-s + 0.307·26-s + 0.335·28-s − 1.67·29-s − 1.50·31-s − 0.859·32-s − 0.291·34-s − 0.181·35-s + 1.59·37-s − 0.569·38-s + 0.303·40-s + 1.21·41-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
| \(L(2)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 + 0.948T + 8T^{2} \) |
| 5 | \( 1 - 5.36T + 125T^{2} \) |
| 13 | \( 1 + 42.9T + 2.19e3T^{2} \) |
| 17 | \( 1 - 60.8T + 4.91e3T^{2} \) |
| 19 | \( 1 - 140.T + 6.85e3T^{2} \) |
| 23 | \( 1 - 91.3T + 1.21e4T^{2} \) |
| 29 | \( 1 + 260.T + 2.43e4T^{2} \) |
| 31 | \( 1 + 259.T + 2.97e4T^{2} \) |
| 37 | \( 1 - 359.T + 5.06e4T^{2} \) |
| 41 | \( 1 - 320.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 92.3T + 7.95e4T^{2} \) |
| 47 | \( 1 + 67.4T + 1.03e5T^{2} \) |
| 53 | \( 1 - 246.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 475.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 799.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 725.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 544.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 580.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 402.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 1.10e3T + 5.71e5T^{2} \) |
| 89 | \( 1 + 1.25e3T + 7.04e5T^{2} \) |
| 97 | \( 1 + 9.99e2T + 9.12e5T^{2} \) |
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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
−9.479264504763567314750885243206, −9.184441188353012851282104559707, −7.68327679839171147352884729232, −7.42069251850984945151730663946, −5.72566311713299537455326859646, −5.27849320457487709593338216840, −4.01275152710589195220119942013, −2.89011072785254005961638972133, −1.34468543103861334978274148012, 0,
1.34468543103861334978274148012, 2.89011072785254005961638972133, 4.01275152710589195220119942013, 5.27849320457487709593338216840, 5.72566311713299537455326859646, 7.42069251850984945151730663946, 7.68327679839171147352884729232, 9.184441188353012851282104559707, 9.479264504763567314750885243206
