L(s) = 1 | + (7.42 − 2.98i)2-s − 6.08·3-s + (46.1 − 44.3i)4-s + (26.7 − 122. i)5-s + (−45.1 + 18.1i)6-s + 422.·7-s + (209. − 466. i)8-s − 691.·9-s + (−165. − 986. i)10-s + 1.74e3i·11-s + (−280. + 269. i)12-s − 898. i·13-s + (3.13e3 − 1.26e3i)14-s + (−163. + 743. i)15-s + (163. − 4.09e3i)16-s + 5.54e3i·17-s + ⋯ |
L(s) = 1 | + (0.927 − 0.373i)2-s − 0.225·3-s + (0.721 − 0.692i)4-s + (0.214 − 0.976i)5-s + (−0.209 + 0.0842i)6-s + 1.23·7-s + (0.410 − 0.912i)8-s − 0.949·9-s + (−0.165 − 0.986i)10-s + 1.31i·11-s + (−0.162 + 0.156i)12-s − 0.409i·13-s + (1.14 − 0.459i)14-s + (−0.0483 + 0.220i)15-s + (0.0398 − 0.999i)16-s + 1.12i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 20 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.522 + 0.852i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(2.16678 - 1.21394i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.16678 - 1.21394i\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-7.42 + 2.98i)T \) |
| 5 | \( 1 + (-26.7 + 122. i)T \) |
good | 3 | \( 1 + 6.08T + 729T^{2} \) |
| 7 | \( 1 - 422.T + 1.17e5T^{2} \) |
| 11 | \( 1 - 1.74e3iT - 1.77e6T^{2} \) |
| 13 | \( 1 + 898. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 5.54e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 - 8.90e3iT - 4.70e7T^{2} \) |
| 23 | \( 1 + 3.74e3T + 1.48e8T^{2} \) |
| 29 | \( 1 - 3.11e4T + 5.94e8T^{2} \) |
| 31 | \( 1 + 2.26e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 + 1.49e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 6.59e3T + 4.75e9T^{2} \) |
| 43 | \( 1 + 8.10e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 3.02e4T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.72e4iT - 2.21e10T^{2} \) |
| 59 | \( 1 - 2.94e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 2.11e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 4.52e5T + 9.04e10T^{2} \) |
| 71 | \( 1 - 2.46e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 4.76e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 6.80e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.03e6T + 3.26e11T^{2} \) |
| 89 | \( 1 + 9.37e5T + 4.96e11T^{2} \) |
| 97 | \( 1 + 9.57e5iT - 8.32e11T^{2} \) |
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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
−16.80631835448183163980627503397, −15.14908879132068362719226865628, −14.17426812136298486542437354410, −12.64242086409372605877947547376, −11.75025564831227674991773138009, −10.21964449180675845290137280382, −8.145102649070898004613984452068, −5.75205046486561198826954102213, −4.49831955723091113826893726466, −1.71317243801573667281940483639,
2.86215581082155319738296639464, 5.14646816707702587630047895423, 6.65608944221055018168778976632, 8.342552924222706634445374978027, 11.10376540398201237671000867360, 11.57380394858278784212913598856, 13.88284374069100350446417057034, 14.25120443504323591235650384466, 15.70302745800937279385864344933, 17.12425682538775627890579672096