L(s) = 1 | + (−0.0341 + 0.0592i)2-s + (−1.15 + 1.29i)3-s + (0.997 + 1.72i)4-s − 2.66·5-s + (−0.0368 − 0.112i)6-s − 0.273·8-s + (−0.329 − 2.98i)9-s + (0.0910 − 0.157i)10-s − 1.59·11-s + (−3.38 − 0.709i)12-s + (−2.62 + 4.54i)13-s + (3.07 − 3.43i)15-s + (−1.98 + 3.43i)16-s + (3.27 − 5.67i)17-s + (0.187 + 0.0824i)18-s + (−0.950 − 1.64i)19-s + ⋯ |
L(s) = 1 | + (−0.0241 + 0.0418i)2-s + (−0.667 + 0.744i)3-s + (0.498 + 0.864i)4-s − 1.19·5-s + (−0.0150 − 0.0459i)6-s − 0.0965·8-s + (−0.109 − 0.993i)9-s + (0.0287 − 0.0498i)10-s − 0.482·11-s + (−0.976 − 0.204i)12-s + (−0.728 + 1.26i)13-s + (0.794 − 0.887i)15-s + (−0.496 + 0.859i)16-s + (0.793 − 1.37i)17-s + (0.0442 + 0.0194i)18-s + (−0.218 − 0.377i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.820 + 0.571i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0786142 - 0.250269i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0786142 - 0.250269i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.15 - 1.29i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.0341 - 0.0592i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + 2.66T + 5T^{2} \) |
| 11 | \( 1 + 1.59T + 11T^{2} \) |
| 13 | \( 1 + (2.62 - 4.54i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-3.27 + 5.67i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.950 + 1.64i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + 3.06T + 23T^{2} \) |
| 29 | \( 1 + (3.19 + 5.53i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (3.35 + 5.81i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.11 + 3.66i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (3.69 - 6.40i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.63 - 9.75i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.89 - 3.29i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (4.44 - 7.70i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.44 - 9.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.35 - 2.35i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.66 - 2.87i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12.3T + 71T^{2} \) |
| 73 | \( 1 + (1.09 - 1.90i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (0.406 - 0.704i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (3.41 + 5.92i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (0.235 + 0.407i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.57 - 4.46i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−11.65773499818108118115395539099, −11.14863950344585558986075909209, −9.855970324062256763655084501576, −9.034146536378036668989905426378, −7.72976839180350129957134398613, −7.28845319176662271387600906272, −6.05820321049261821225996271836, −4.63024953227338370651395872032, −3.97534928549390843888308378425, −2.75066725306946431375117667409,
0.17084859042079694861917161173, 1.81420424477388257931260640107, 3.45515916790747738433068196661, 5.12623086149280330557653863970, 5.73974949286270062210375115917, 6.94393676661131074972063152127, 7.67765753481683185184999548834, 8.426446250196352621879262448245, 10.26252034947111585047396445891, 10.52806223787066819209099927869