| L(s) = 1 | + (3.23 + 2.35i)2-s + (4.94 + 15.2i)4-s + (−17.6 − 24.2i)5-s + (−154. + 50.2i)7-s + (−19.7 + 60.8i)8-s − 120. i·10-s + (−185. + 356. i)11-s + (524. − 721. i)13-s + (−618. − 201. i)14-s + (−207. + 150. i)16-s + (1.03e3 − 752. i)17-s + (1.49e3 + 484. i)19-s + (282. − 388. i)20-s + (−1.43e3 + 716. i)22-s − 2.91e3i·23-s + ⋯ |
| L(s) = 1 | + (0.572 + 0.415i)2-s + (0.154 + 0.475i)4-s + (−0.315 − 0.434i)5-s + (−1.19 + 0.387i)7-s + (−0.109 + 0.336i)8-s − 0.379i·10-s + (−0.461 + 0.887i)11-s + (0.860 − 1.18i)13-s + (−0.843 − 0.274i)14-s + (−0.202 + 0.146i)16-s + (0.869 − 0.631i)17-s + (0.947 + 0.307i)19-s + (0.157 − 0.217i)20-s + (−0.632 + 0.315i)22-s − 1.14i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 198 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.640 + 0.768i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.716999163\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.716999163\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-3.23 - 2.35i)T \) |
| 3 | \( 1 \) |
| 11 | \( 1 + (185. - 356. i)T \) |
| good | 5 | \( 1 + (17.6 + 24.2i)T + (-965. + 2.97e3i)T^{2} \) |
| 7 | \( 1 + (154. - 50.2i)T + (1.35e4 - 9.87e3i)T^{2} \) |
| 13 | \( 1 + (-524. + 721. i)T + (-1.14e5 - 3.53e5i)T^{2} \) |
| 17 | \( 1 + (-1.03e3 + 752. i)T + (4.38e5 - 1.35e6i)T^{2} \) |
| 19 | \( 1 + (-1.49e3 - 484. i)T + (2.00e6 + 1.45e6i)T^{2} \) |
| 23 | \( 1 + 2.91e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (957. + 2.94e3i)T + (-1.65e7 + 1.20e7i)T^{2} \) |
| 31 | \( 1 + (4.14e3 + 3.00e3i)T + (8.84e6 + 2.72e7i)T^{2} \) |
| 37 | \( 1 + (2.02e3 + 6.24e3i)T + (-5.61e7 + 4.07e7i)T^{2} \) |
| 41 | \( 1 + (-5.77e3 + 1.77e4i)T + (-9.37e7 - 6.80e7i)T^{2} \) |
| 43 | \( 1 - 2.36e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (1.11e4 + 3.61e3i)T + (1.85e8 + 1.34e8i)T^{2} \) |
| 53 | \( 1 + (-2.27e4 + 3.13e4i)T + (-1.29e8 - 3.97e8i)T^{2} \) |
| 59 | \( 1 + (2.60e4 - 8.46e3i)T + (5.78e8 - 4.20e8i)T^{2} \) |
| 61 | \( 1 + (-2.03e4 - 2.80e4i)T + (-2.60e8 + 8.03e8i)T^{2} \) |
| 67 | \( 1 - 6.01e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (1.79e4 + 2.46e4i)T + (-5.57e8 + 1.71e9i)T^{2} \) |
| 73 | \( 1 + (4.95e4 - 1.61e4i)T + (1.67e9 - 1.21e9i)T^{2} \) |
| 79 | \( 1 + (-2.61e4 + 3.60e4i)T + (-9.50e8 - 2.92e9i)T^{2} \) |
| 83 | \( 1 + (2.78e3 - 2.02e3i)T + (1.21e9 - 3.74e9i)T^{2} \) |
| 89 | \( 1 - 5.39e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.27e4 - 4.56e4i)T + (2.65e9 + 8.16e9i)T^{2} \) |
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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
−11.82671664037011812257489008020, −10.39814513688733177449694742498, −9.472312942220792897932423106697, −8.240400629377992413290209459673, −7.32533866994283951558308036091, −6.07161554674738603292020080489, −5.20330192664007725328666781363, −3.79228450234418202481364772249, −2.69488924630588054080543810243, −0.47121010829217646894250977511,
1.26274096029428721153195107584, 3.22864323668501253141620116586, 3.65589003859058381142586039909, 5.41523434959343438827504020546, 6.45348207944697501972094404119, 7.42288105384333002671111101355, 8.947873157762150358605230182401, 9.935369737864347617586443136193, 10.95198775715000007002199221939, 11.63325725754235517500838152140
