| L(s) = 1 | + (−1 + i)2-s + (−1.22 − 1.22i)3-s − 2i·4-s + 2.44·6-s + (1.87 − 1.87i)7-s + (2 + 2i)8-s + 2.99i·9-s − 3.02·11-s + (−2.44 + 2.44i)12-s + (2.99 + 2.99i)13-s + 3.74i·14-s − 4·16-s + (−13.0 + 13.0i)17-s + (−2.99 − 2.99i)18-s − 29.5i·19-s + ⋯ |
| L(s) = 1 | + (−0.5 + 0.5i)2-s + (−0.408 − 0.408i)3-s − 0.5i·4-s + 0.408·6-s + (0.267 − 0.267i)7-s + (0.250 + 0.250i)8-s + 0.333i·9-s − 0.274·11-s + (−0.204 + 0.204i)12-s + (0.230 + 0.230i)13-s + 0.267i·14-s − 0.250·16-s + (−0.768 + 0.768i)17-s + (−0.166 − 0.166i)18-s − 1.55i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.945 + 0.326i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.1928410505\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1928410505\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1 - i)T \) |
| 3 | \( 1 + (1.22 + 1.22i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (-1.87 + 1.87i)T \) |
| good | 11 | \( 1 + 3.02T + 121T^{2} \) |
| 13 | \( 1 + (-2.99 - 2.99i)T + 169iT^{2} \) |
| 17 | \( 1 + (13.0 - 13.0i)T - 289iT^{2} \) |
| 19 | \( 1 + 29.5iT - 361T^{2} \) |
| 23 | \( 1 + (2.34 + 2.34i)T + 529iT^{2} \) |
| 29 | \( 1 + 24.4iT - 841T^{2} \) |
| 31 | \( 1 - 46.7T + 961T^{2} \) |
| 37 | \( 1 + (30.6 - 30.6i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 11.9T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-0.402 - 0.402i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + (46.4 - 46.4i)T - 2.20e3iT^{2} \) |
| 53 | \( 1 + (11.9 + 11.9i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + 32.2iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 78.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + (-17.3 + 17.3i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 43.2T + 5.04e3T^{2} \) |
| 73 | \( 1 + (39.4 + 39.4i)T + 5.32e3iT^{2} \) |
| 79 | \( 1 - 0.100iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-15.9 - 15.9i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 - 91.4iT - 7.92e3T^{2} \) |
| 97 | \( 1 + (14.1 - 14.1i)T - 9.40e3iT^{2} \) |
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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
−9.235966595922737638951555173358, −8.374988157185156791577640653324, −7.74258890433478920551884299114, −6.68404337008437399622300915381, −6.30434661960384223205011595589, −5.06739507515371623201066332805, −4.33396417364312182655247485169, −2.66175969289697407909667409084, −1.39156135656590926572776644340, −0.07839459542403006080630070633,
1.44542964164582666185956038866, 2.73655004573546611868548314149, 3.82197564350753191727200375160, 4.83626124270879928785832243091, 5.75361764027247218077279338850, 6.76853049906667146609660510411, 7.80231199039843185659750945065, 8.576081139404064795197023750203, 9.346729739638173171665380000034, 10.24162737219950724343847493324
