L(s) = 1 | + (1.22 − 1.22i)3-s + (−3.71 + 3.71i)5-s − 5.63·7-s − 2.99i·9-s + (5.26 + 5.26i)11-s + (−1.74 − 1.74i)13-s + 9.11i·15-s − 1.43·17-s + (22.1 − 22.1i)19-s + (−6.90 + 6.90i)21-s − 2.27·23-s − 2.67i·25-s + (−3.67 − 3.67i)27-s + (−35.9 − 35.9i)29-s − 13.3i·31-s + ⋯ |
L(s) = 1 | + (0.408 − 0.408i)3-s + (−0.743 + 0.743i)5-s − 0.805·7-s − 0.333i·9-s + (0.478 + 0.478i)11-s + (−0.133 − 0.133i)13-s + 0.607i·15-s − 0.0846·17-s + (1.16 − 1.16i)19-s + (−0.328 + 0.328i)21-s − 0.0988·23-s − 0.106i·25-s + (−0.136 − 0.136i)27-s + (−1.23 − 1.23i)29-s − 0.429i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 768 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.130 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.108142267\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.108142267\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.22 + 1.22i)T \) |
good | 5 | \( 1 + (3.71 - 3.71i)T - 25iT^{2} \) |
| 7 | \( 1 + 5.63T + 49T^{2} \) |
| 11 | \( 1 + (-5.26 - 5.26i)T + 121iT^{2} \) |
| 13 | \( 1 + (1.74 + 1.74i)T + 169iT^{2} \) |
| 17 | \( 1 + 1.43T + 289T^{2} \) |
| 19 | \( 1 + (-22.1 + 22.1i)T - 361iT^{2} \) |
| 23 | \( 1 + 2.27T + 529T^{2} \) |
| 29 | \( 1 + (35.9 + 35.9i)T + 841iT^{2} \) |
| 31 | \( 1 + 13.3iT - 961T^{2} \) |
| 37 | \( 1 + (-36.1 + 36.1i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 + 63.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (46.5 + 46.5i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 - 61.0iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-49.0 + 49.0i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (-41.6 - 41.6i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (40.4 + 40.4i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-36.0 + 36.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 - 8.24T + 5.04e3T^{2} \) |
| 73 | \( 1 + 122. iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 93.0iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-21.6 + 21.6i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 119. iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 115.T + 9.40e3T^{2} \) |
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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
−9.632181215007680127730984579798, −9.252928814100859298426659853402, −7.943315358836796676758392746386, −7.24937429045329309424411461960, −6.70332577738029003862381100865, −5.54431292352321551894414424136, −4.07403336381606802700473530665, −3.30113589915089394444296524349, −2.25182306419685054093174443774, −0.39018200297421202012403544070,
1.27051183719230011023459309062, 3.10790873810493648026079826302, 3.79238726567923521410438893122, 4.81993051508227964560275382800, 5.86097705429613116639179096869, 6.96909258728244142164284134944, 7.990334142183319709246122075852, 8.626902857517836441462810081495, 9.536203903302006472479244336199, 10.07052769043194348004524443491