L(s) = 1 | + (1 − i)2-s − 1.73i·3-s − 2i·4-s + (0.866 − 1.5i)5-s + (−1.73 − 1.73i)6-s + (1.73 − 2i)7-s + (−2 − 2i)8-s − 2.99·9-s + (−0.633 − 2.36i)10-s + (2.5 + 4.33i)11-s − 3.46·12-s + (0.866 + 1.5i)13-s + (−0.267 − 3.73i)14-s + (−2.59 − 1.49i)15-s − 4·16-s + (1.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s − 0.999i·3-s − i·4-s + (0.387 − 0.670i)5-s + (−0.707 − 0.707i)6-s + (0.654 − 0.755i)7-s + (−0.707 − 0.707i)8-s − 0.999·9-s + (−0.200 − 0.748i)10-s + (0.753 + 1.30i)11-s − 1.00·12-s + (0.240 + 0.416i)13-s + (−0.0716 − 0.997i)14-s + (−0.670 − 0.387i)15-s − 16-s + (0.363 + 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.853 + 0.520i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.610661 - 2.17476i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.610661 - 2.17476i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 3 | \( 1 + 1.73iT \) |
| 7 | \( 1 + (-1.73 + 2i)T \) |
good | 5 | \( 1 + (-0.866 + 1.5i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (-2.5 - 4.33i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-0.866 - 1.5i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 - 0.866i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.5 - 0.866i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-6.06 - 3.5i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (6.06 + 3.5i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 6.92T + 31T^{2} \) |
| 37 | \( 1 + (-0.866 + 0.5i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-10.5 + 6.06i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (4.5 - 7.79i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 3.46T + 47T^{2} \) |
| 53 | \( 1 + (9.52 + 5.5i)T + (26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + 6.92iT - 59T^{2} \) |
| 61 | \( 1 - 3.46T + 61T^{2} \) |
| 67 | \( 1 + 8T + 67T^{2} \) |
| 71 | \( 1 - 10iT - 71T^{2} \) |
| 73 | \( 1 + (7.5 + 4.33i)T + (36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 - 6iT - 79T^{2} \) |
| 83 | \( 1 + (-13.5 - 7.79i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-7.5 + 4.33i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.5 - 0.866i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.98061195536271840498321492519, −9.620626581793326268685949025080, −9.025190350941085321027385135747, −7.60362909598465610715550719740, −6.85588749814962395249424105277, −5.71248863452681977192840752830, −4.78228879906470452476985319236, −3.71683860070410738945971181652, −1.94873601377448382464476098208, −1.27609857696236131307261105900,
2.72160982628742062208172612566, 3.55249505503508545433117141973, 4.79828964259493927283058134538, 5.71144359971032062863901817971, 6.30093137896911086764559836516, 7.64287110195845455600571975348, 8.839497247834557555156406670810, 9.061458436412093636200179357806, 10.76277471541343374099936443170, 11.11869377660800236781804366240