L(s) = 1 | + (−1.5 + 0.866i)3-s + (1 + 1.73i)5-s + 7-s + (1.5 − 2.59i)9-s − 16·11-s + (13.5 + 7.79i)13-s + (−3 − 1.73i)15-s + (−11 − 19.0i)17-s − 19·19-s + (−1.5 + 0.866i)21-s + (20 − 34.6i)23-s + (10.5 − 18.1i)25-s + 5.19i·27-s + (15 + 8.66i)29-s + 50.2i·31-s + ⋯ |
L(s) = 1 | + (−0.5 + 0.288i)3-s + (0.200 + 0.346i)5-s + 0.142·7-s + (0.166 − 0.288i)9-s − 1.45·11-s + (1.03 + 0.599i)13-s + (−0.200 − 0.115i)15-s + (−0.647 − 1.12i)17-s − 19-s + (−0.0714 + 0.0412i)21-s + (0.869 − 1.50i)23-s + (0.419 − 0.727i)25-s + 0.192i·27-s + (0.517 + 0.298i)29-s + 1.62i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 912 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.412 + 0.910i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.053594361\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053594361\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.5 - 0.866i)T \) |
| 19 | \( 1 + 19T \) |
good | 5 | \( 1 + (-1 - 1.73i)T + (-12.5 + 21.6i)T^{2} \) |
| 7 | \( 1 - T + 49T^{2} \) |
| 11 | \( 1 + 16T + 121T^{2} \) |
| 13 | \( 1 + (-13.5 - 7.79i)T + (84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (11 + 19.0i)T + (-144.5 + 250. i)T^{2} \) |
| 23 | \( 1 + (-20 + 34.6i)T + (-264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (-15 - 8.66i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 50.2iT - 961T^{2} \) |
| 37 | \( 1 - 15.5iT - 1.36e3T^{2} \) |
| 41 | \( 1 + (12 - 6.92i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (24.5 + 42.4i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + (-23 + 39.8i)T + (-1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (42 + 24.2i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-57 + 32.9i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-48.5 + 84.0i)T + (-1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.5 - 12.9i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 + (-42 + 24.2i)T + (2.52e3 - 4.36e3i)T^{2} \) |
| 73 | \( 1 + (17.5 + 30.3i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-76.5 + 44.1i)T + (3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 - 146T + 6.88e3T^{2} \) |
| 89 | \( 1 + (33 + 19.0i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (-54 + 31.1i)T + (4.70e3 - 8.14e3i)T^{2} \) |
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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
−9.974818738230425908094484805558, −8.767915940042164497238343812642, −8.297527240441737650583372561174, −6.76004682635692444977232427449, −6.58195562829461375867208906795, −5.13329655323861776094303914536, −4.67535189614147769874615993908, −3.24806116852241526100197985956, −2.17894078228272399717872110227, −0.40289497913648998223260979410,
1.13856745598346454833444355645, 2.40105091743757672843814327720, 3.76998426192481984713727362576, 4.95156242445396215273711041273, 5.70980782495832438522797120028, 6.44528218927577536328470491133, 7.67798130070173116229662946795, 8.224230624011749101885159893770, 9.162395086499320923245610504022, 10.24663271245405403042240815131