L(s) = 1 | + (0.746 + 1.20i)2-s + (−2.58 + 1.63i)3-s + (−0.884 + 1.79i)4-s + (0.800 + 0.0590i)5-s + (−3.89 − 1.87i)6-s + (−0.732 + 0.808i)7-s + (−2.81 + 0.278i)8-s + (2.70 − 5.72i)9-s + (0.527 + 1.00i)10-s + (−4.27 + 0.105i)11-s + (−0.650 − 6.07i)12-s + (−1.71 + 5.18i)13-s + (−1.51 − 0.276i)14-s + (−2.16 + 1.15i)15-s + (−2.43 − 3.17i)16-s + (3.01 − 5.63i)17-s + ⋯ |
L(s) = 1 | + (0.528 + 0.849i)2-s + (−1.49 + 0.943i)3-s + (−0.442 + 0.896i)4-s + (0.358 + 0.0264i)5-s + (−1.58 − 0.766i)6-s + (−0.276 + 0.305i)7-s + (−0.995 + 0.0983i)8-s + (0.902 − 1.90i)9-s + (0.166 + 0.318i)10-s + (−1.28 + 0.0316i)11-s + (−0.187 − 1.75i)12-s + (−0.475 + 1.43i)13-s + (−0.405 − 0.0737i)14-s + (−0.558 + 0.298i)15-s + (−0.609 − 0.793i)16-s + (0.730 − 1.36i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0111 + 0.999i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0111 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.246315 - 0.249083i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.246315 - 0.249083i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.746 - 1.20i)T \) |
good | 3 | \( 1 + (2.58 - 1.63i)T + (1.28 - 2.71i)T^{2} \) |
| 5 | \( 1 + (-0.800 - 0.0590i)T + (4.94 + 0.733i)T^{2} \) |
| 7 | \( 1 + (0.732 - 0.808i)T + (-0.686 - 6.96i)T^{2} \) |
| 11 | \( 1 + (4.27 - 0.105i)T + (10.9 - 0.539i)T^{2} \) |
| 13 | \( 1 + (1.71 - 5.18i)T + (-10.4 - 7.74i)T^{2} \) |
| 17 | \( 1 + (-3.01 + 5.63i)T + (-9.44 - 14.1i)T^{2} \) |
| 19 | \( 1 + (-0.394 + 3.19i)T + (-18.4 - 4.61i)T^{2} \) |
| 23 | \( 1 + (-3.23 - 5.39i)T + (-10.8 + 20.2i)T^{2} \) |
| 29 | \( 1 + (-4.43 + 1.70i)T + (21.4 - 19.4i)T^{2} \) |
| 31 | \( 1 + (2.40 + 1.60i)T + (11.8 + 28.6i)T^{2} \) |
| 37 | \( 1 + (1.72 - 6.21i)T + (-31.7 - 19.0i)T^{2} \) |
| 41 | \( 1 + (2.23 - 0.331i)T + (39.2 - 11.9i)T^{2} \) |
| 43 | \( 1 + (1.20 + 5.37i)T + (-38.8 + 18.3i)T^{2} \) |
| 47 | \( 1 + (-2.59 - 0.256i)T + (46.0 + 9.16i)T^{2} \) |
| 53 | \( 1 + (4.40 + 1.70i)T + (39.2 + 35.5i)T^{2} \) |
| 59 | \( 1 + (13.2 - 4.38i)T + (47.3 - 35.1i)T^{2} \) |
| 61 | \( 1 + (6.22 - 1.08i)T + (57.4 - 20.5i)T^{2} \) |
| 67 | \( 1 + (0.714 - 1.01i)T + (-22.5 - 63.0i)T^{2} \) |
| 71 | \( 1 + (8.12 + 2.90i)T + (54.8 + 45.0i)T^{2} \) |
| 73 | \( 1 + (10.4 + 11.5i)T + (-7.15 + 72.6i)T^{2} \) |
| 79 | \( 1 + (0.894 + 0.734i)T + (15.4 + 77.4i)T^{2} \) |
| 83 | \( 1 + (-3.34 - 12.0i)T + (-71.1 + 42.6i)T^{2} \) |
| 89 | \( 1 + (7.37 - 12.3i)T + (-41.9 - 78.4i)T^{2} \) |
| 97 | \( 1 + (-0.773 - 3.88i)T + (-89.6 + 37.1i)T^{2} \) |
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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
−11.87676610893718152508108405597, −10.73937147246325305168986374210, −9.612480416165297372804109069607, −9.263746309826237454542188444516, −7.58490749442119092480380773752, −6.73873889230872134414068798584, −5.78590868558298716197173052488, −5.10069778554613409587479481267, −4.49135603728626638221445380738, −2.99815557440256797276783220047,
0.20705709924827310987296775101, 1.61302045247201522565784168270, 3.05159205586540760910197920669, 4.73110995516875745308650565132, 5.66760090648372103640581254436, 6.00513878957744572016132363003, 7.32576247056621337163457890142, 8.280784332570718544324577911738, 10.17574467459623292105117539472, 10.34715076030964507970321614431