| L(s) = 1 | + (−1.21 − 0.727i)2-s + (0.941 + 1.76i)4-s + (0.0465 − 0.127i)5-s + (−0.252 + 1.42i)7-s + (0.142 − 2.82i)8-s + (−0.149 + 0.121i)10-s + (0.771 + 2.11i)11-s + (0.634 + 0.755i)13-s + (1.34 − 1.55i)14-s + (−2.22 + 3.32i)16-s + (0.439 + 0.760i)17-s + (−5.20 − 3.00i)19-s + (0.269 − 0.0382i)20-s + (0.606 − 3.13i)22-s + (0.748 + 4.24i)23-s + ⋯ |
| L(s) = 1 | + (−0.857 − 0.514i)2-s + (0.470 + 0.882i)4-s + (0.0208 − 0.0572i)5-s + (−0.0952 + 0.540i)7-s + (0.0503 − 0.998i)8-s + (−0.0473 + 0.0383i)10-s + (0.232 + 0.638i)11-s + (0.175 + 0.209i)13-s + (0.359 − 0.414i)14-s + (−0.557 + 0.830i)16-s + (0.106 + 0.184i)17-s + (−1.19 − 0.689i)19-s + (0.0602 − 0.00855i)20-s + (0.129 − 0.667i)22-s + (0.156 + 0.885i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 648 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.542 - 0.840i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.722158 + 0.393318i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.722158 + 0.393318i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.21 + 0.727i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-0.0465 + 0.127i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (0.252 - 1.42i)T + (-6.57 - 2.39i)T^{2} \) |
| 11 | \( 1 + (-0.771 - 2.11i)T + (-8.42 + 7.07i)T^{2} \) |
| 13 | \( 1 + (-0.634 - 0.755i)T + (-2.25 + 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.439 - 0.760i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (5.20 + 3.00i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.748 - 4.24i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-0.146 + 0.174i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (-1.15 - 6.56i)T + (-29.1 + 10.6i)T^{2} \) |
| 37 | \( 1 + (9.25 - 5.34i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.19 + 1.00i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (1.11 + 3.05i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.36 - 7.74i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 5.08iT - 53T^{2} \) |
| 59 | \( 1 + (3.15 - 8.68i)T + (-45.1 - 37.9i)T^{2} \) |
| 61 | \( 1 + (-12.3 - 2.18i)T + (57.3 + 20.8i)T^{2} \) |
| 67 | \( 1 + (-5.90 - 7.03i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (-5.93 - 10.2i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.88 + 8.45i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-5.49 - 4.60i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (0.867 - 1.03i)T + (-14.4 - 81.7i)T^{2} \) |
| 89 | \( 1 + (-3.19 + 5.54i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.779 + 0.283i)T + (74.3 - 62.3i)T^{2} \) |
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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
−10.64146398535215118815754912357, −9.799547814081469611634196672696, −8.927996936374971552654086002466, −8.443682216832287513478885023233, −7.18512016066421254832712079530, −6.57820293296790597341536708814, −5.15788728672899322778432941684, −3.89761208459480289121513372664, −2.70625046916265510198027436168, −1.50967213485140057611538261171,
0.60906915524438472641187869827, 2.24454834461668267785162340062, 3.81097645237479055379002453825, 5.12531206501935378236189498499, 6.25422428686827245276313142810, 6.81591505653377486188402857645, 7.981418822411820925040912578312, 8.546302679515282822306752714184, 9.476120283594416933001247587216, 10.51095174918528676038639611411
