| L(s) = 1 | + 3.29i·3-s − 5.90i·5-s + (6.86 − 1.36i)7-s − 1.82·9-s + 5.68·11-s − 10.7i·13-s + 19.4·15-s + 16.6i·17-s − 11.4i·19-s + (4.48 + 22.5i)21-s + 33.1·23-s − 9.82·25-s + 23.5i·27-s + 18.9·29-s + 59.6i·31-s + ⋯ |
| L(s) = 1 | + 1.09i·3-s − 1.18i·5-s + (0.980 − 0.194i)7-s − 0.203·9-s + 0.517·11-s − 0.830i·13-s + 1.29·15-s + 0.981i·17-s − 0.603i·19-s + (0.213 + 1.07i)21-s + 1.44·23-s − 0.393·25-s + 0.874i·27-s + 0.654·29-s + 1.92i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.980 - 0.194i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.78611 + 0.175575i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.78611 + 0.175575i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + (-6.86 + 1.36i)T \) |
| good | 3 | \( 1 - 3.29iT - 9T^{2} \) |
| 5 | \( 1 + 5.90iT - 25T^{2} \) |
| 11 | \( 1 - 5.68T + 121T^{2} \) |
| 13 | \( 1 + 10.7iT - 169T^{2} \) |
| 17 | \( 1 - 16.6iT - 289T^{2} \) |
| 19 | \( 1 + 11.4iT - 361T^{2} \) |
| 23 | \( 1 - 33.1T + 529T^{2} \) |
| 29 | \( 1 - 18.9T + 841T^{2} \) |
| 31 | \( 1 - 59.6iT - 961T^{2} \) |
| 37 | \( 1 + 45.5T + 1.36e3T^{2} \) |
| 41 | \( 1 + 68.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 33.1T + 1.84e3T^{2} \) |
| 47 | \( 1 + 54.2iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 11.3T + 2.80e3T^{2} \) |
| 59 | \( 1 - 22.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 64.9iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 110.T + 4.48e3T^{2} \) |
| 71 | \( 1 + 9.01T + 5.04e3T^{2} \) |
| 73 | \( 1 + 14.6iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 102.T + 6.24e3T^{2} \) |
| 83 | \( 1 - 87.7iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 95.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 140. iT - 9.40e3T^{2} \) |
| 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
−12.08038789981058277109686169979, −10.82814331986357256874813689463, −10.27880221582396154766469792335, −8.839125549345802638588440477718, −8.622843189558860845415817626986, −7.07772853206287145670865544180, −5.24005910405019410314661704747, −4.80611795932329029011245354114, −3.60044221971035916559545385279, −1.29070391300834712007505454307,
1.49073216232455192430767826889, 2.79291122625720929048959554779, 4.54816336694364954124770915688, 6.16004487288975080955394904317, 7.03349203649112421352006881723, 7.67410684659757463155550071102, 8.920107047863437450562342378929, 10.17346398502307308533296191561, 11.47058460614251549766343056541, 11.67333312043476849638214711662
