| L(s) = 1 | − i·2-s − 4-s + 2.82i·5-s + 2.64i·7-s + i·8-s + 2.82·10-s + (−2.64 − 2i)11-s − 5.65·13-s + 2.64·14-s + 16-s + 7.48·17-s + 2.82·19-s − 2.82i·20-s + (−2 + 2.64i)22-s − 5.29·23-s + ⋯ |
| L(s) = 1 | − 0.707i·2-s − 0.5·4-s + 1.26i·5-s + 0.999i·7-s + 0.353i·8-s + 0.894·10-s + (−0.797 − 0.603i)11-s − 1.56·13-s + 0.707·14-s + 0.250·16-s + 1.81·17-s + 0.648·19-s − 0.632i·20-s + (−0.426 + 0.564i)22-s − 1.10·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1386 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.797 - 0.603i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.4722568893\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4722568893\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + iT \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| 11 | \( 1 + (2.64 + 2i)T \) |
| good | 5 | \( 1 - 2.82iT - 5T^{2} \) |
| 13 | \( 1 + 5.65T + 13T^{2} \) |
| 17 | \( 1 - 7.48T + 17T^{2} \) |
| 19 | \( 1 - 2.82T + 19T^{2} \) |
| 23 | \( 1 + 5.29T + 23T^{2} \) |
| 29 | \( 1 + 6iT - 29T^{2} \) |
| 31 | \( 1 - 7.48iT - 31T^{2} \) |
| 37 | \( 1 + 10T + 37T^{2} \) |
| 41 | \( 1 + 7.48T + 41T^{2} \) |
| 43 | \( 1 - 5.29iT - 43T^{2} \) |
| 47 | \( 1 + 2.82iT - 47T^{2} \) |
| 53 | \( 1 + 10.5T + 53T^{2} \) |
| 59 | \( 1 + 11.3iT - 59T^{2} \) |
| 61 | \( 1 + 61T^{2} \) |
| 67 | \( 1 + 12T + 67T^{2} \) |
| 71 | \( 1 - 5.29T + 71T^{2} \) |
| 73 | \( 1 + 8.48T + 73T^{2} \) |
| 79 | \( 1 - 5.29iT - 79T^{2} \) |
| 83 | \( 1 - 7.48T + 83T^{2} \) |
| 89 | \( 1 - 89T^{2} \) |
| 97 | \( 1 - 14.9iT - 97T^{2} \) |
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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
−10.09669496215821428039755174822, −9.401686263182821348983154215457, −8.169949197017066115416012235192, −7.67714167758711153880047443999, −6.60337203540486524920213565897, −5.56327798520500882630702373741, −5.00457509483208379631948442463, −3.31852628811714322182090010487, −2.97095095659529107701584463775, −1.94655416429873027856829605912,
0.18621788739744660289619857472, 1.57547708011735467768548439087, 3.29741973019958222012984350847, 4.47456853154522467368132903435, 5.06640466423529800535371073276, 5.70037924180315207213443857438, 7.16165871334272008085242863845, 7.57064932629527577447142864097, 8.205823849302990508640932378096, 9.275122191738918103155248665340
