| L(s) = 1 | + 6.10·2-s + 5.30·4-s + 25·5-s − 241.·7-s − 163.·8-s + 152.·10-s − 121·11-s + 1.16e3·13-s − 1.47e3·14-s − 1.16e3·16-s − 635.·17-s − 1.60e3·19-s + 132.·20-s − 739.·22-s + 3.01e3·23-s + 625·25-s + 7.10e3·26-s − 1.28e3·28-s + 6.39e3·29-s − 3.03e3·31-s − 1.90e3·32-s − 3.87e3·34-s − 6.04e3·35-s − 9.54e3·37-s − 9.79e3·38-s − 4.07e3·40-s + 1.37e4·41-s + ⋯ |
| L(s) = 1 | + 1.07·2-s + 0.165·4-s + 0.447·5-s − 1.86·7-s − 0.900·8-s + 0.482·10-s − 0.301·11-s + 1.90·13-s − 2.01·14-s − 1.13·16-s − 0.532·17-s − 1.01·19-s + 0.0741·20-s − 0.325·22-s + 1.18·23-s + 0.200·25-s + 2.06·26-s − 0.309·28-s + 1.41·29-s − 0.566·31-s − 0.328·32-s − 0.575·34-s − 0.834·35-s − 1.14·37-s − 1.09·38-s − 0.402·40-s + 1.27·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 495 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.592032876\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.592032876\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 11 | \( 1 + 121T \) |
| good | 2 | \( 1 - 6.10T + 32T^{2} \) |
| 7 | \( 1 + 241.T + 1.68e4T^{2} \) |
| 13 | \( 1 - 1.16e3T + 3.71e5T^{2} \) |
| 17 | \( 1 + 635.T + 1.41e6T^{2} \) |
| 19 | \( 1 + 1.60e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.01e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 6.39e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.03e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.54e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.37e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 1.35e4T + 1.47e8T^{2} \) |
| 47 | \( 1 - 556.T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.69e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 2.26e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 3.45e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 5.74e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 8.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 5.09e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 1.11e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 1.91e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 8.27e4T + 8.58e9T^{2} \) |
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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
−10.28127363760703587508901683783, −9.061285855273429686721577728358, −8.758697725723081745173225622466, −6.85863061543018703464031371629, −6.26081682740133126793157045300, −5.61633710363562711252896672787, −4.25277305284783523587433341749, −3.43458573880553709289767548144, −2.58145996050106165310042834278, −0.67420081854836685543846049623,
0.67420081854836685543846049623, 2.58145996050106165310042834278, 3.43458573880553709289767548144, 4.25277305284783523587433341749, 5.61633710363562711252896672787, 6.26081682740133126793157045300, 6.85863061543018703464031371629, 8.758697725723081745173225622466, 9.061285855273429686721577728358, 10.28127363760703587508901683783
