L(s) = 1 | + (−0.866 + 0.5i)2-s + (0.499 − 0.866i)4-s − 2.44·5-s + (2.62 + 0.358i)7-s + 0.999i·8-s + (2.12 − 1.22i)10-s + 4.24i·11-s + (0.621 − 0.358i)13-s + (−2.44 + i)14-s + (−0.5 − 0.866i)16-s + (1.22 + 2.12i)17-s + (−4.24 − 2.44i)19-s + (−1.22 + 2.12i)20-s + (−2.12 − 3.67i)22-s − 6i·23-s + ⋯ |
L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s − 1.09·5-s + (0.990 + 0.135i)7-s + 0.353i·8-s + (0.670 − 0.387i)10-s + 1.27i·11-s + (0.172 − 0.0994i)13-s + (−0.654 + 0.267i)14-s + (−0.125 − 0.216i)16-s + (0.297 + 0.514i)17-s + (−0.973 − 0.561i)19-s + (−0.273 + 0.474i)20-s + (−0.452 − 0.783i)22-s − 1.25i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1134 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.7259708638\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7259708638\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.866 - 0.5i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.62 - 0.358i)T \) |
good | 5 | \( 1 + 2.44T + 5T^{2} \) |
| 11 | \( 1 - 4.24iT - 11T^{2} \) |
| 13 | \( 1 + (-0.621 + 0.358i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 2.12i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (4.24 + 2.44i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + (1.52 + 0.878i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-7.86 - 4.54i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.62 - 4.54i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (1.22 + 2.12i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.5 - 6.06i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (-6.42 - 11.1i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (12.5 - 7.24i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.22 - 2.12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.62 - 2.09i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (6.74 - 11.6i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 12.7iT - 71T^{2} \) |
| 73 | \( 1 + (4.75 - 2.74i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.378 + 0.655i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.64 + 13.2i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + (-1.52 + 2.63i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.74 - 1.58i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.21698481251350304598760315996, −9.036648413777908328379987162562, −8.315238223053032249897836983324, −7.79758987565618829567842821922, −7.00250731722213165445418355142, −6.09956971921012416143742984877, −4.66686488667979718083177962763, −4.37894277570026584570274558987, −2.69711651946668094716947870572, −1.39158279958045387003945535096,
0.41643311164452117719419900668, 1.82692531123717948946328657958, 3.32341555593014793211934535159, 4.00438830800858746094855578152, 5.15606086410710904688881908614, 6.27235635858340727894941641271, 7.43604167819905483371487095240, 8.046821508202452002026694226548, 8.511218728904305149509579467709, 9.452703139465288854775530426952