| L(s) = 1 | + (1.24 − 0.677i)2-s + (−0.740 + 0.671i)3-s + (1.08 − 1.68i)4-s + (−0.0647 − 0.258i)5-s + (−0.464 + 1.33i)6-s + (0.00672 + 0.00551i)7-s + (0.204 − 2.82i)8-s + (0.0980 − 0.995i)9-s + (−0.255 − 0.277i)10-s + (2.13 − 1.00i)11-s + (0.327 + 1.97i)12-s + (−1.81 + 1.09i)13-s + (0.0120 + 0.00229i)14-s + (0.221 + 0.148i)15-s + (−1.65 − 3.64i)16-s + (3.61 − 2.41i)17-s + ⋯ |
| L(s) = 1 | + (0.877 − 0.478i)2-s + (−0.427 + 0.387i)3-s + (0.541 − 0.840i)4-s + (−0.0289 − 0.115i)5-s + (−0.189 + 0.545i)6-s + (0.00254 + 0.00208i)7-s + (0.0722 − 0.997i)8-s + (0.0326 − 0.331i)9-s + (−0.0807 − 0.0876i)10-s + (0.642 − 0.304i)11-s + (0.0945 + 0.569i)12-s + (−0.504 + 0.302i)13-s + (0.00323 + 0.000613i)14-s + (0.0572 + 0.0382i)15-s + (−0.414 − 0.910i)16-s + (0.877 − 0.586i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.285 + 0.958i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.80454 - 1.34471i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80454 - 1.34471i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-1.24 + 0.677i)T \) |
| 3 | \( 1 + (0.740 - 0.671i)T \) |
| good | 5 | \( 1 + (0.0647 + 0.258i)T + (-4.40 + 2.35i)T^{2} \) |
| 7 | \( 1 + (-0.00672 - 0.00551i)T + (1.36 + 6.86i)T^{2} \) |
| 11 | \( 1 + (-2.13 + 1.00i)T + (6.97 - 8.50i)T^{2} \) |
| 13 | \( 1 + (1.81 - 1.09i)T + (6.12 - 11.4i)T^{2} \) |
| 17 | \( 1 + (-3.61 + 2.41i)T + (6.50 - 15.7i)T^{2} \) |
| 19 | \( 1 + (-5.02 + 6.77i)T + (-5.51 - 18.1i)T^{2} \) |
| 23 | \( 1 + (7.46 - 2.26i)T + (19.1 - 12.7i)T^{2} \) |
| 29 | \( 1 + (-4.67 + 1.67i)T + (22.4 - 18.3i)T^{2} \) |
| 31 | \( 1 + (-0.898 - 2.16i)T + (-21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.557 - 3.76i)T + (-35.4 - 10.7i)T^{2} \) |
| 41 | \( 1 + (4.24 + 2.26i)T + (22.7 + 34.0i)T^{2} \) |
| 43 | \( 1 + (5.42 - 5.98i)T + (-4.21 - 42.7i)T^{2} \) |
| 47 | \( 1 + (-0.491 + 2.47i)T + (-43.4 - 17.9i)T^{2} \) |
| 53 | \( 1 + (-9.64 - 3.44i)T + (40.9 + 33.6i)T^{2} \) |
| 59 | \( 1 + (8.65 + 5.18i)T + (27.8 + 52.0i)T^{2} \) |
| 61 | \( 1 + (-13.6 - 0.672i)T + (60.7 + 5.97i)T^{2} \) |
| 67 | \( 1 + (0.787 - 16.0i)T + (-66.6 - 6.56i)T^{2} \) |
| 71 | \( 1 + (-1.19 + 0.117i)T + (69.6 - 13.8i)T^{2} \) |
| 73 | \( 1 + (-4.11 + 3.37i)T + (14.2 - 71.5i)T^{2} \) |
| 79 | \( 1 + (7.93 - 1.57i)T + (72.9 - 30.2i)T^{2} \) |
| 83 | \( 1 + (-1.99 - 13.4i)T + (-79.4 + 24.0i)T^{2} \) |
| 89 | \( 1 + (13.3 + 4.04i)T + (74.0 + 49.4i)T^{2} \) |
| 97 | \( 1 + (-2.26 + 0.936i)T + (68.5 - 68.5i)T^{2} \) |
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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
−10.05135653733025538994578400743, −9.781828332953312126533884491396, −8.610270822088680889825869098954, −7.19342896970476316668494287960, −6.49232213704240653230605324467, −5.37865790658268125857806505631, −4.77669022525791500089452311950, −3.70701825678165159783277894040, −2.67855504730580172096997406242, −0.992616541361562332116767010018,
1.69879261051321427801557213757, 3.18936896136333049636155747168, 4.16964073649841219493874218199, 5.33075905660868876568418613550, 5.99096755161575652476120144523, 6.89539334853591686716821524723, 7.71824254746895923969466971724, 8.385451977245350016900013859389, 9.830398352622801672152371333734, 10.56002050550177907258803096506
