L(s) = 1 | + 1.50·2-s + (−2.08 − 2.15i)3-s − 1.72·4-s + (−4.79 − 1.41i)5-s + (−3.14 − 3.24i)6-s − 2.64i·7-s − 8.63·8-s + (−0.288 + 8.99i)9-s + (−7.22 − 2.13i)10-s − 0.491i·11-s + (3.60 + 3.72i)12-s − 21.9i·13-s − 3.98i·14-s + (6.95 + 13.2i)15-s − 6.11·16-s + 4.43·17-s + ⋯ |
L(s) = 1 | + 0.753·2-s + (−0.695 − 0.718i)3-s − 0.431·4-s + (−0.958 − 0.283i)5-s + (−0.524 − 0.541i)6-s − 0.377i·7-s − 1.07·8-s + (−0.0321 + 0.999i)9-s + (−0.722 − 0.213i)10-s − 0.0446i·11-s + (0.300 + 0.310i)12-s − 1.68i·13-s − 0.284i·14-s + (0.463 + 0.886i)15-s − 0.381·16-s + 0.260·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 105 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.886 + 0.463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.159395 - 0.648790i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159395 - 0.648790i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (2.08 + 2.15i)T \) |
| 5 | \( 1 + (4.79 + 1.41i)T \) |
| 7 | \( 1 + 2.64iT \) |
good | 2 | \( 1 - 1.50T + 4T^{2} \) |
| 11 | \( 1 + 0.491iT - 121T^{2} \) |
| 13 | \( 1 + 21.9iT - 169T^{2} \) |
| 17 | \( 1 - 4.43T + 289T^{2} \) |
| 19 | \( 1 - 14.4T + 361T^{2} \) |
| 23 | \( 1 + 19.9T + 529T^{2} \) |
| 29 | \( 1 + 35.6iT - 841T^{2} \) |
| 31 | \( 1 + 45.5T + 961T^{2} \) |
| 37 | \( 1 - 58.9iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 45.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.94iT - 1.84e3T^{2} \) |
| 47 | \( 1 + 41.5T + 2.20e3T^{2} \) |
| 53 | \( 1 - 18.4T + 2.80e3T^{2} \) |
| 59 | \( 1 + 9.08iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 46.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 92.9iT - 4.48e3T^{2} \) |
| 71 | \( 1 + 37.1iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 4.57iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 80.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 14.0T + 6.88e3T^{2} \) |
| 89 | \( 1 + 155. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 124. iT - 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−12.97656629448821278673651060361, −12.26536905232749634584300170603, −11.39234181967562530548225719254, −10.08404337450807163110656384638, −8.323276535041873070258121752185, −7.49045116280383043460131199961, −5.88203019786772423452411658480, −4.90047884720251738114473859162, −3.45414372476262000931837399621, −0.43459637001000193295654526509,
3.54817719758390702713834180149, 4.45420228136607896505420761449, 5.62857946796845684327920926501, 6.97815030694815061116847401894, 8.756696584728712548785238271852, 9.672183611492709321450929843258, 11.19703388399283144385299497805, 11.86575203311258089924776125413, 12.69454013531909457265754249303, 14.30721599248308020919402269401