L(s) = 1 | + (1.87 − 0.684i)2-s + (−2.95 + 0.493i)3-s + (3.06 − 2.57i)4-s + (−5.22 + 2.95i)6-s + (4.00 − 6.92i)8-s + (8.51 − 2.92i)9-s + (3.75 − 21.3i)11-s + (−7.79 + 9.12i)12-s + (2.77 − 15.7i)16-s + (6.74 + 11.6i)17-s + (14 − 11.3i)18-s + (11.3 − 19.5i)19-s + (−7.51 − 42.6i)22-s + (−8.41 + 22.4i)24-s + (−23.4 + 8.55i)25-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (−0.986 + 0.164i)3-s + (0.766 − 0.642i)4-s + (−0.870 + 0.491i)6-s + (0.500 − 0.866i)8-s + (0.945 − 0.324i)9-s + (0.341 − 1.93i)11-s + (−0.649 + 0.760i)12-s + (0.173 − 0.984i)16-s + (0.396 + 0.687i)17-s + (0.777 − 0.628i)18-s + (0.595 − 1.03i)19-s + (−0.341 − 1.93i)22-s + (−0.350 + 0.936i)24-s + (−0.939 + 0.342i)25-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.260 + 0.965i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.59450 - 1.22173i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.59450 - 1.22173i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.87 + 0.684i)T \) |
| 3 | \( 1 + (2.95 - 0.493i)T \) |
good | 5 | \( 1 + (23.4 - 8.55i)T^{2} \) |
| 7 | \( 1 + (-8.50 - 48.2i)T^{2} \) |
| 11 | \( 1 + (-3.75 + 21.3i)T + (-113. - 41.3i)T^{2} \) |
| 13 | \( 1 + (-129. - 108. i)T^{2} \) |
| 17 | \( 1 + (-6.74 - 11.6i)T + (-144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-11.3 + 19.5i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (-91.8 + 520. i)T^{2} \) |
| 29 | \( 1 + (-644. + 540. i)T^{2} \) |
| 31 | \( 1 + (-166. + 946. i)T^{2} \) |
| 37 | \( 1 + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-70.3 - 25.5i)T + (1.28e3 + 1.08e3i)T^{2} \) |
| 43 | \( 1 + (7.32 - 41.5i)T + (-1.73e3 - 632. i)T^{2} \) |
| 47 | \( 1 + (-383. - 2.17e3i)T^{2} \) |
| 53 | \( 1 - 2.80e3T^{2} \) |
| 59 | \( 1 + (-12.0 - 68.2i)T + (-3.27e3 + 1.19e3i)T^{2} \) |
| 61 | \( 1 + (-646. - 3.66e3i)T^{2} \) |
| 67 | \( 1 + (116. + 42.3i)T + (3.43e3 + 2.88e3i)T^{2} \) |
| 71 | \( 1 + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (60.9 - 105. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-4.78e3 + 4.01e3i)T^{2} \) |
| 83 | \( 1 + (-32.8 + 11.9i)T + (5.27e3 - 4.42e3i)T^{2} \) |
| 89 | \( 1 + (-80.5 + 139. i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-6.43 + 36.5i)T + (-8.84e3 - 3.21e3i)T^{2} \) |
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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
−11.65972687256174244270309086966, −11.26805608198800647013623350923, −10.38848975617351672764852289686, −9.197525598061013969901254015233, −7.54470515852636803140320636141, −6.17455136733857796510638406767, −5.71966907216962021590665691176, −4.39051893026324788575227025587, −3.20140842967109006895557688265, −1.02003777088971115717093615898,
1.94405684849745852986105479750, 3.98692914527065820882820828697, 4.97365103142143727610703544375, 5.96038150433376893810299221870, 7.09920316775304796037931319633, 7.68685486289544847985872335933, 9.623428725169525713575537991998, 10.53404770108246590583201256712, 11.93516008460080964689790489157, 12.07338937263999786952334107377