| L(s) = 1 | + (−0.183 + 2.10i)2-s + (−0.448 − 0.0791i)4-s + (−1.81 − 4.66i)5-s + (−2.07 − 1.45i)7-s + (−1.93 + 7.22i)8-s + (10.1 − 2.95i)10-s + (−1.77 − 0.646i)11-s + (0.975 + 11.1i)13-s + (3.43 − 4.09i)14-s + (−16.5 − 6.02i)16-s + (5.53 + 20.6i)17-s + (−26.0 + 15.0i)19-s + (0.444 + 2.23i)20-s + (1.68 − 3.61i)22-s + (−1.45 + 1.01i)23-s + ⋯ |
| L(s) = 1 | + (−0.0919 + 1.05i)2-s + (−0.112 − 0.0197i)4-s + (−0.362 − 0.932i)5-s + (−0.296 − 0.207i)7-s + (−0.242 + 0.903i)8-s + (1.01 − 0.295i)10-s + (−0.161 − 0.0587i)11-s + (0.0750 + 0.857i)13-s + (0.245 − 0.292i)14-s + (−1.03 − 0.376i)16-s + (0.325 + 1.21i)17-s + (−1.37 + 0.792i)19-s + (0.0222 + 0.111i)20-s + (0.0766 − 0.164i)22-s + (−0.0631 + 0.0442i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0379i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.999 - 0.0379i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.0168390 + 0.886695i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0168390 + 0.886695i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.81 + 4.66i)T \) |
| good | 2 | \( 1 + (0.183 - 2.10i)T + (-3.93 - 0.694i)T^{2} \) |
| 7 | \( 1 + (2.07 + 1.45i)T + (16.7 + 46.0i)T^{2} \) |
| 11 | \( 1 + (1.77 + 0.646i)T + (92.6 + 77.7i)T^{2} \) |
| 13 | \( 1 + (-0.975 - 11.1i)T + (-166. + 29.3i)T^{2} \) |
| 17 | \( 1 + (-5.53 - 20.6i)T + (-250. + 144.5i)T^{2} \) |
| 19 | \( 1 + (26.0 - 15.0i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (1.45 - 1.01i)T + (180. - 497. i)T^{2} \) |
| 29 | \( 1 + (5.76 + 6.87i)T + (-146. + 828. i)T^{2} \) |
| 31 | \( 1 + (7.07 - 40.1i)T + (-903. - 328. i)T^{2} \) |
| 37 | \( 1 + (0.867 + 3.23i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + (-36.9 - 30.9i)T + (291. + 1.65e3i)T^{2} \) |
| 43 | \( 1 + (-15.2 - 32.6i)T + (-1.18e3 + 1.41e3i)T^{2} \) |
| 47 | \( 1 + (31.6 + 22.1i)T + (755. + 2.07e3i)T^{2} \) |
| 53 | \( 1 + (7.77 + 7.77i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (37.8 + 104. i)T + (-2.66e3 + 2.23e3i)T^{2} \) |
| 61 | \( 1 + (-1.75 - 9.93i)T + (-3.49e3 + 1.27e3i)T^{2} \) |
| 67 | \( 1 + (-95.6 + 8.36i)T + (4.42e3 - 779. i)T^{2} \) |
| 71 | \( 1 + (45.9 - 79.5i)T + (-2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (-3.86 + 14.4i)T + (-4.61e3 - 2.66e3i)T^{2} \) |
| 79 | \( 1 + (73.7 + 87.9i)T + (-1.08e3 + 6.14e3i)T^{2} \) |
| 83 | \( 1 + (-123. - 10.7i)T + (6.78e3 + 1.19e3i)T^{2} \) |
| 89 | \( 1 + (-21.5 + 12.4i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (35.8 - 16.6i)T + (6.04e3 - 7.20e3i)T^{2} \) |
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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
−11.53712515188381447636738695298, −10.57149189470148094492896031852, −9.353940091951641654817578019341, −8.402840690746496850791102219927, −7.938167947930386579090517065725, −6.71298862167711979797209547939, −5.99465297664323685111295752512, −4.85391554028974583874231316506, −3.75814449960615865594129576373, −1.80895403903723972992196724352,
0.37154766309342469256656863170, 2.36781484734871932964823145949, 3.06496857864546377458577593531, 4.23775643031235146068614240801, 5.86598026718561792596261438277, 6.87246076427163985864097965615, 7.72691994224194941421645746735, 9.096570675643798649551040641168, 9.958447709160042473408406052713, 10.80314122315170984093492170177
