L(s) = 1 | + i·2-s − 4-s + (1 + i)5-s − i·8-s + i·9-s + (−1 + i)10-s + 16-s − 18-s + (−1 − i)20-s + i·25-s + (−1 − i)29-s + i·32-s − i·36-s + (1 + i)37-s + (1 − i)40-s + (−1 + i)41-s + ⋯ |
L(s) = 1 | + i·2-s − 4-s + (1 + i)5-s − i·8-s + i·9-s + (−1 + i)10-s + 16-s − 18-s + (−1 − i)20-s + i·25-s + (−1 − i)29-s + i·32-s − i·36-s + (1 + i)37-s + (1 − i)40-s + (−1 + i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.615 - 0.788i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.082126591\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.082126591\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 - iT \) |
| 17 | \( 1 \) |
good | 3 | \( 1 - iT^{2} \) |
| 5 | \( 1 + (-1 - i)T + iT^{2} \) |
| 7 | \( 1 + iT^{2} \) |
| 11 | \( 1 + iT^{2} \) |
| 13 | \( 1 + T^{2} \) |
| 19 | \( 1 + T^{2} \) |
| 23 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (1 + i)T + iT^{2} \) |
| 31 | \( 1 - iT^{2} \) |
| 37 | \( 1 + (-1 - i)T + iT^{2} \) |
| 41 | \( 1 + (1 - i)T - iT^{2} \) |
| 43 | \( 1 + T^{2} \) |
| 47 | \( 1 - T^{2} \) |
| 53 | \( 1 - T^{2} \) |
| 59 | \( 1 + T^{2} \) |
| 61 | \( 1 + (-1 + i)T - iT^{2} \) |
| 67 | \( 1 - T^{2} \) |
| 71 | \( 1 - iT^{2} \) |
| 73 | \( 1 + (1 + i)T + iT^{2} \) |
| 79 | \( 1 + iT^{2} \) |
| 83 | \( 1 + T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (-1 - i)T + iT^{2} \) |
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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
−10.00970584902981600579973287175, −9.607415386083401094956271544170, −8.439657804706711083837858769858, −7.72879367723669114854597092930, −6.87669512574765925648394428030, −6.16740746409478676644620606513, −5.42975100836414362453355487544, −4.50164390692390560834591818762, −3.18467872154381935110746738526, −1.97516299636307899317861391759,
1.06916387095544160553343369755, 2.13342109937984989800887388856, 3.41317344429636947129541897105, 4.38084617078799069393105239350, 5.37946989256413027767597702369, 5.97990528087817090924620174529, 7.29810120254522126109350699028, 8.595052282691261720801964921958, 9.062988826822614822111352980273, 9.652845597817256245731247013466