L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.866 + 1.5i)3-s + (0.499 + 0.866i)4-s − 1.73i·6-s + (−3.46 − 2i)7-s − 0.999i·8-s + (−1.5 + 2.59i)9-s + (−1.5 + 2.59i)11-s + (−0.866 + 1.49i)12-s + (−3.46 + 2i)13-s + (1.99 + 3.46i)14-s + (−0.5 + 0.866i)16-s + 3i·17-s + (2.59 − 1.5i)18-s + 4·19-s + ⋯ |
L(s) = 1 | + (−0.612 − 0.353i)2-s + (0.499 + 0.866i)3-s + (0.249 + 0.433i)4-s − 0.707i·6-s + (−1.30 − 0.755i)7-s − 0.353i·8-s + (−0.5 + 0.866i)9-s + (−0.452 + 0.783i)11-s + (−0.250 + 0.433i)12-s + (−0.960 + 0.554i)13-s + (0.534 + 0.925i)14-s + (−0.125 + 0.216i)16-s + 0.727i·17-s + (0.612 − 0.353i)18-s + 0.917·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.803 - 0.595i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.159931 + 0.484084i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.159931 + 0.484084i\) |
\(L(\frac{3}{2})\) |
|
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 | \( 1 + (-0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
good | 7 | \( 1 + (3.46 + 2i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.5 - 2.59i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (3.46 - 2i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 3iT - 17T^{2} \) |
| 19 | \( 1 - 4T + 19T^{2} \) |
| 23 | \( 1 + (5.19 - 3i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3 - 5.19i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (4 + 6.92i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 8iT - 37T^{2} \) |
| 41 | \( 1 + (-3 - 5.19i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.866 - 0.5i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-10.3 - 6i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 53T^{2} \) |
| 59 | \( 1 + (4.5 + 7.79i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (4 - 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.46 + 2i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 6T + 71T^{2} \) |
| 73 | \( 1 - 14iT - 73T^{2} \) |
| 79 | \( 1 + (-4 + 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-7.79 - 4.5i)T + (41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 - 9T + 89T^{2} \) |
| 97 | \( 1 + (6.06 + 3.5i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.09133355156240697934871354930, −10.28858308953453197578912390397, −9.582797124663055277520035863773, −9.278848499138418737448832083855, −7.74581612966021127894014292688, −7.24747657830854254685185883887, −5.77776783729059045135028451623, −4.31245964003600668225425439618, −3.48061156615724392144904788543, −2.24585239739276046473037093398,
0.33534494210286195735546655434, 2.41769554535967638132457069447, 3.25125020821996948147486049315, 5.43957702713819270948110051389, 6.19409333673517089030868964211, 7.18046604038872709637514970533, 7.937856997430255495321165883651, 8.935810971597588098737479582533, 9.555150374161498330389564556330, 10.47156958604286775645411237143