L(s) = 1 | + (−0.5 + 0.866i)2-s + (−0.499 − 0.866i)4-s + (−0.822 + 1.42i)5-s + 2.64·7-s + 0.999·8-s + (−0.822 − 1.42i)10-s + (0.822 + 1.42i)11-s + 0.645·13-s + (−1.32 + 2.29i)14-s + (−0.5 + 0.866i)16-s + (0.822 + 1.42i)17-s + (−1 + 1.73i)19-s + 1.64·20-s − 1.64·22-s + (−4.64 + 8.04i)23-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.368 + 0.637i)5-s + 0.999·7-s + 0.353·8-s + (−0.260 − 0.450i)10-s + (0.248 + 0.429i)11-s + 0.179·13-s + (−0.353 + 0.612i)14-s + (−0.125 + 0.216i)16-s + (0.199 + 0.345i)17-s + (−0.229 + 0.397i)19-s + 0.368·20-s − 0.350·22-s + (−0.968 + 1.67i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0633 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.761368 + 0.811216i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.761368 + 0.811216i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 - 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64T \) |
good | 5 | \( 1 + (0.822 - 1.42i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.822 - 1.42i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 - 0.645T + 13T^{2} \) |
| 17 | \( 1 + (-0.822 - 1.42i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1 - 1.73i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.64 - 8.04i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 7.64T + 29T^{2} \) |
| 31 | \( 1 + (0.322 + 0.559i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.96 - 3.40i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + 4.93T + 41T^{2} \) |
| 43 | \( 1 - 5T + 43T^{2} \) |
| 47 | \( 1 + (-5.46 + 9.47i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (3 + 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (6.82 + 11.8i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (6.32 - 10.9i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.14 + 7.18i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 10.3T + 71T^{2} \) |
| 73 | \( 1 + (-5.29 - 9.16i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-7.61 + 13.1i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 2.70T + 83T^{2} \) |
| 89 | \( 1 + (-5.46 + 9.47i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 7.58T + 97T^{2} \) |
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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
−11.50102337308517826149283513178, −10.61592598813957507019349822464, −9.786290337793231095702881694129, −8.599583532178804732287486178455, −7.82857788504580043170789860364, −7.06112646755552965656900715277, −5.95613676277264001187766085596, −4.83614649200238246715473806049, −3.61261227028754427283149458135, −1.71275111514828229095086366608,
0.939764235495284517461314573234, 2.53757378193520524351596309838, 4.15374233038599652949557590217, 4.89003474636887829065215335840, 6.35843007795052429720274206397, 7.77695139122755657889418922965, 8.454783021816057202428604028600, 9.131875808979930665446662371854, 10.46020033958906419998642294484, 11.02200609709205114638738197604