| L(s) = 1 | + (0.300 − 1.38i)2-s + (−1.98 − 0.976i)3-s + (−1.81 − 0.831i)4-s + (0.453 − 1.33i)5-s + (−1.94 + 2.44i)6-s + (−1.09 − 2.40i)7-s + (−1.69 + 2.26i)8-s + (1.14 + 1.48i)9-s + (−1.71 − 1.02i)10-s + (−2.00 − 0.131i)11-s + (2.79 + 3.42i)12-s + (1.44 + 0.287i)13-s + (−3.65 + 0.795i)14-s + (−2.20 + 2.20i)15-s + (2.61 + 3.02i)16-s + (0.246 − 0.919i)17-s + ⋯ |
| L(s) = 1 | + (0.212 − 0.977i)2-s + (−1.14 − 0.563i)3-s + (−0.909 − 0.415i)4-s + (0.202 − 0.598i)5-s + (−0.794 + 0.997i)6-s + (−0.415 − 0.909i)7-s + (−0.599 + 0.800i)8-s + (0.380 + 0.495i)9-s + (−0.541 − 0.325i)10-s + (−0.606 − 0.0397i)11-s + (0.805 + 0.987i)12-s + (0.400 + 0.0797i)13-s + (−0.977 + 0.212i)14-s + (−0.569 + 0.569i)15-s + (0.654 + 0.756i)16-s + (0.0597 − 0.222i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 448 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.154 - 0.988i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.273142 + 0.319104i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.273142 + 0.319104i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.300 + 1.38i)T \) |
| 7 | \( 1 + (1.09 + 2.40i)T \) |
| good | 3 | \( 1 + (1.98 + 0.976i)T + (1.82 + 2.38i)T^{2} \) |
| 5 | \( 1 + (-0.453 + 1.33i)T + (-3.96 - 3.04i)T^{2} \) |
| 11 | \( 1 + (2.00 + 0.131i)T + (10.9 + 1.43i)T^{2} \) |
| 13 | \( 1 + (-1.44 - 0.287i)T + (12.0 + 4.97i)T^{2} \) |
| 17 | \( 1 + (-0.246 + 0.919i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (0.608 + 0.533i)T + (2.47 + 18.8i)T^{2} \) |
| 23 | \( 1 + (3.58 - 2.74i)T + (5.95 - 22.2i)T^{2} \) |
| 29 | \( 1 + (-0.289 - 0.433i)T + (-11.0 + 26.7i)T^{2} \) |
| 31 | \( 1 + (-2.93 + 1.69i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (11.3 + 3.85i)T + (29.3 + 22.5i)T^{2} \) |
| 41 | \( 1 + (-9.40 - 3.89i)T + (28.9 + 28.9i)T^{2} \) |
| 43 | \( 1 + (5.75 + 3.84i)T + (16.4 + 39.7i)T^{2} \) |
| 47 | \( 1 + (-1.88 + 0.504i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (6.00 + 0.393i)T + (52.5 + 6.91i)T^{2} \) |
| 59 | \( 1 + (7.50 + 8.55i)T + (-7.70 + 58.4i)T^{2} \) |
| 61 | \( 1 + (0.454 + 6.93i)T + (-60.4 + 7.96i)T^{2} \) |
| 67 | \( 1 + (3.73 + 1.84i)T + (40.7 + 53.1i)T^{2} \) |
| 71 | \( 1 + (1.19 + 2.88i)T + (-50.2 + 50.2i)T^{2} \) |
| 73 | \( 1 + (8.79 - 1.15i)T + (70.5 - 18.8i)T^{2} \) |
| 79 | \( 1 + (-0.493 - 1.84i)T + (-68.4 + 39.5i)T^{2} \) |
| 83 | \( 1 + (-0.998 + 5.01i)T + (-76.6 - 31.7i)T^{2} \) |
| 89 | \( 1 + (0.308 - 2.34i)T + (-85.9 - 23.0i)T^{2} \) |
| 97 | \( 1 + 15.3iT - 97T^{2} \) |
| 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
−10.71040015089793842616172020268, −9.862723825825026746543297616987, −8.873582318046896094974851229057, −7.63565650285803618240075382954, −6.43372446218313037367168044317, −5.52008002351335841709043557158, −4.64589816720230313740247346370, −3.36371172428359779001795245633, −1.54407509844635990623258228553, −0.29018105643229852234827678176,
2.92057422316781311048682684842, 4.37439890680202939932797772029, 5.38554012822484695066977394300, 6.05941001293870760490868123631, 6.71615356103057028393054067327, 8.058868867582253955468560658021, 8.960959677952060613964600905756, 10.12799635583265046713630298286, 10.62048142291364282234642130059, 11.89583092398541583669761915207
