| L(s) = 1 | + (−1.82 + 0.820i)2-s + (−2.97 + 0.420i)3-s + (2.65 − 2.99i)4-s + (3.62 − 3.44i)5-s + (5.07 − 3.20i)6-s + (8.40 − 8.40i)7-s + (−2.38 + 7.63i)8-s + (8.64 − 2.49i)9-s + (−3.78 + 9.25i)10-s + (−15.5 − 11.3i)11-s + (−6.62 + 10.0i)12-s + (−1.80 + 11.3i)13-s + (−8.43 + 22.2i)14-s + (−9.32 + 11.7i)15-s + (−1.92 − 15.8i)16-s + (5.37 + 10.5i)17-s + ⋯ |
| L(s) = 1 | + (−0.911 + 0.410i)2-s + (−0.990 + 0.140i)3-s + (0.663 − 0.748i)4-s + (0.725 − 0.688i)5-s + (0.845 − 0.534i)6-s + (1.20 − 1.20i)7-s + (−0.297 + 0.954i)8-s + (0.960 − 0.277i)9-s + (−0.378 + 0.925i)10-s + (−1.41 − 1.02i)11-s + (−0.551 + 0.833i)12-s + (−0.138 + 0.876i)13-s + (−0.602 + 1.58i)14-s + (−0.621 + 0.783i)15-s + (−0.120 − 0.992i)16-s + (0.316 + 0.620i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 300 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.133 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.478797 - 0.547382i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.478797 - 0.547382i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.82 - 0.820i)T \) |
| 3 | \( 1 + (2.97 - 0.420i)T \) |
| 5 | \( 1 + (-3.62 + 3.44i)T \) |
| good | 7 | \( 1 + (-8.40 + 8.40i)T - 49iT^{2} \) |
| 11 | \( 1 + (15.5 + 11.3i)T + (37.3 + 115. i)T^{2} \) |
| 13 | \( 1 + (1.80 - 11.3i)T + (-160. - 52.2i)T^{2} \) |
| 17 | \( 1 + (-5.37 - 10.5i)T + (-169. + 233. i)T^{2} \) |
| 19 | \( 1 + (-1.22 + 3.78i)T + (-292. - 212. i)T^{2} \) |
| 23 | \( 1 + (2.57 - 0.408i)T + (503. - 163. i)T^{2} \) |
| 29 | \( 1 + (15.6 + 48.3i)T + (-680. + 494. i)T^{2} \) |
| 31 | \( 1 + (8.55 + 2.77i)T + (777. + 564. i)T^{2} \) |
| 37 | \( 1 + (-3.43 - 0.544i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (-31.5 - 43.4i)T + (-519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (27.2 + 27.2i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (9.52 - 18.7i)T + (-1.29e3 - 1.78e3i)T^{2} \) |
| 53 | \( 1 + (-19.1 + 37.5i)T + (-1.65e3 - 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-20.4 - 28.1i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (75.2 + 54.6i)T + (1.14e3 + 3.53e3i)T^{2} \) |
| 67 | \( 1 + (8.79 + 17.2i)T + (-2.63e3 + 3.63e3i)T^{2} \) |
| 71 | \( 1 + (24.2 + 74.5i)T + (-4.07e3 + 2.96e3i)T^{2} \) |
| 73 | \( 1 + (43.8 - 6.94i)T + (5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (8.03 + 24.7i)T + (-5.04e3 + 3.66e3i)T^{2} \) |
| 83 | \( 1 + (-29.6 - 58.1i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 + (-70.0 - 50.8i)T + (2.44e3 + 7.53e3i)T^{2} \) |
| 97 | \( 1 + (21.1 - 41.5i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−10.98945918913744291962334212014, −10.40104202053944984321555796187, −9.537708418589799340203200882747, −8.237309703041659729598063032657, −7.57300596963857700642434302249, −6.26393612670528966379648350354, −5.40552641473105036386880306877, −4.49071120501346243976189513137, −1.77371585945381869698131491348, −0.53938666593790973683040965884,
1.71661327436881458704629256101, 2.70973618319842140698196389314, 5.06264581312277471886494064607, 5.72855003898565638046501353837, 7.19858351611089318242314243663, 7.80542085944353452009448179272, 9.124840946581257312490688609598, 10.23180356806970927882167422045, 10.67220834667076487332649371719, 11.58490113206285485539400676738
