| L(s) = 1 | + (−0.807 + 1.58i)2-s + (0.0974 − 0.0154i)3-s + (0.492 + 0.678i)4-s + (−0.0542 + 0.166i)6-s + (−2.29 − 0.362i)7-s + (−8.49 + 1.34i)8-s + (−8.55 + 2.77i)9-s + (10.4 − 3.35i)11-s + (0.0584 + 0.0584i)12-s + (−10.0 − 5.13i)13-s + (2.42 − 3.33i)14-s + (3.69 − 11.3i)16-s + (−23.3 + 11.9i)17-s + (2.50 − 15.7i)18-s + (−6.52 + 8.98i)19-s + ⋯ |
| L(s) = 1 | + (−0.403 + 0.792i)2-s + (0.0324 − 0.00514i)3-s + (0.123 + 0.169i)4-s + (−0.00903 + 0.0278i)6-s + (−0.327 − 0.0518i)7-s + (−1.06 + 0.168i)8-s + (−0.950 + 0.308i)9-s + (0.952 − 0.305i)11-s + (0.00487 + 0.00487i)12-s + (−0.775 − 0.395i)13-s + (0.173 − 0.238i)14-s + (0.230 − 0.710i)16-s + (−1.37 + 0.700i)17-s + (0.138 − 0.877i)18-s + (−0.343 + 0.472i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 275 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.753 + 0.657i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.121684 - 0.324488i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.121684 - 0.324488i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 11 | \( 1 + (-10.4 + 3.35i)T \) |
| good | 2 | \( 1 + (0.807 - 1.58i)T + (-2.35 - 3.23i)T^{2} \) |
| 3 | \( 1 + (-0.0974 + 0.0154i)T + (8.55 - 2.78i)T^{2} \) |
| 7 | \( 1 + (2.29 + 0.362i)T + (46.6 + 15.1i)T^{2} \) |
| 13 | \( 1 + (10.0 + 5.13i)T + (99.3 + 136. i)T^{2} \) |
| 17 | \( 1 + (23.3 - 11.9i)T + (169. - 233. i)T^{2} \) |
| 19 | \( 1 + (6.52 - 8.98i)T + (-111. - 343. i)T^{2} \) |
| 23 | \( 1 + (2.20 - 2.20i)T - 529iT^{2} \) |
| 29 | \( 1 + (-9.68 - 13.3i)T + (-259. + 799. i)T^{2} \) |
| 31 | \( 1 + (7.66 + 23.5i)T + (-777. + 564. i)T^{2} \) |
| 37 | \( 1 + (22.0 + 3.49i)T + (1.30e3 + 423. i)T^{2} \) |
| 41 | \( 1 + (42.0 + 30.5i)T + (519. + 1.59e3i)T^{2} \) |
| 43 | \( 1 + (-22.3 + 22.3i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-0.432 - 2.73i)T + (-2.10e3 + 682. i)T^{2} \) |
| 53 | \( 1 + (-72.8 - 37.1i)T + (1.65e3 + 2.27e3i)T^{2} \) |
| 59 | \( 1 + (-27.6 - 38.0i)T + (-1.07e3 + 3.31e3i)T^{2} \) |
| 61 | \( 1 + (15.8 - 48.7i)T + (-3.01e3 - 2.18e3i)T^{2} \) |
| 67 | \( 1 + (1.56 + 1.56i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 + (33.2 - 102. i)T + (-4.07e3 - 2.96e3i)T^{2} \) |
| 73 | \( 1 + (12.5 - 79.5i)T + (-5.06e3 - 1.64e3i)T^{2} \) |
| 79 | \( 1 + (80.4 - 26.1i)T + (5.04e3 - 3.66e3i)T^{2} \) |
| 83 | \( 1 + (6.21 + 12.1i)T + (-4.04e3 + 5.57e3i)T^{2} \) |
| 89 | \( 1 - 92.0iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (-69.8 + 137. i)T + (-5.53e3 - 7.61e3i)T^{2} \) |
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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
−12.13582401872246772956356503156, −11.39078561968928740225336255523, −10.26916183439253456060553310201, −8.924148395155406536830752794135, −8.526947804044345223564421432572, −7.33023905593249253746522167522, −6.44657634107026555074501044087, −5.55358978812780102959036283910, −3.84559317705187666350951763711, −2.46760926704918464305393328443,
0.17943066971209059622340841293, 2.04399548361928286162898022766, 3.17530599781004685189187714073, 4.74643092453690753507496046146, 6.24945340397381553487921999702, 6.92060196894208769935262570431, 8.695390665205046872496444100056, 9.251983252327226063777947614116, 10.10732516277269549820065684372, 11.28386697810573181453332789053
