L(s) = 1 | + (−1.59 + 1.59i)2-s + (−1.36 + 1.06i)3-s − 3.05i·4-s + (−0.812 − 2.08i)5-s + (0.465 − 3.86i)6-s + (1.68 + 1.68i)8-s + (0.711 − 2.91i)9-s + (4.60 + 2.02i)10-s − 4.48i·11-s + (3.27 + 4.16i)12-s + (−1.08 + 1.08i)13-s + (3.33 + 1.96i)15-s + 0.762·16-s + (−1.49 + 1.49i)17-s + (3.50 + 5.76i)18-s + 4.22i·19-s + ⋯ |
L(s) = 1 | + (−1.12 + 1.12i)2-s + (−0.786 + 0.617i)3-s − 1.52i·4-s + (−0.363 − 0.931i)5-s + (0.189 − 1.57i)6-s + (0.595 + 0.595i)8-s + (0.237 − 0.971i)9-s + (1.45 + 0.639i)10-s − 1.35i·11-s + (0.944 + 1.20i)12-s + (−0.300 + 0.300i)13-s + (0.861 + 0.508i)15-s + 0.190·16-s + (−0.363 + 0.363i)17-s + (0.825 + 1.35i)18-s + 0.969i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 + 0.0203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 735 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 + 0.0203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.00215393 - 0.211738i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.00215393 - 0.211738i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.36 - 1.06i)T \) |
| 5 | \( 1 + (0.812 + 2.08i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (1.59 - 1.59i)T - 2iT^{2} \) |
| 11 | \( 1 + 4.48iT - 11T^{2} \) |
| 13 | \( 1 + (1.08 - 1.08i)T - 13iT^{2} \) |
| 17 | \( 1 + (1.49 - 1.49i)T - 17iT^{2} \) |
| 19 | \( 1 - 4.22iT - 19T^{2} \) |
| 23 | \( 1 + (2.29 + 2.29i)T + 23iT^{2} \) |
| 29 | \( 1 + 1.69T + 29T^{2} \) |
| 31 | \( 1 + 1.06T + 31T^{2} \) |
| 37 | \( 1 + (-4.21 - 4.21i)T + 37iT^{2} \) |
| 41 | \( 1 - 5.84iT - 41T^{2} \) |
| 43 | \( 1 + (-2.00 + 2.00i)T - 43iT^{2} \) |
| 47 | \( 1 + (3.73 - 3.73i)T - 47iT^{2} \) |
| 53 | \( 1 + (-6.11 - 6.11i)T + 53iT^{2} \) |
| 59 | \( 1 - 4.70T + 59T^{2} \) |
| 61 | \( 1 + 7.77T + 61T^{2} \) |
| 67 | \( 1 + (0.416 + 0.416i)T + 67iT^{2} \) |
| 71 | \( 1 - 4.66iT - 71T^{2} \) |
| 73 | \( 1 + (-3.08 + 3.08i)T - 73iT^{2} \) |
| 79 | \( 1 + 6.67iT - 79T^{2} \) |
| 83 | \( 1 + (11.0 + 11.0i)T + 83iT^{2} \) |
| 89 | \( 1 - 3.51T + 89T^{2} \) |
| 97 | \( 1 + (5.60 + 5.60i)T + 97iT^{2} \) |
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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
−10.53234321587847045139934739251, −9.733850666573119166050823629923, −8.948361585605209635669357001281, −8.359325417387891698769747074851, −7.52031971572325244507058423836, −6.24672727031290064891597519621, −5.85482278057862467791895258157, −4.76707834331825568475519934513, −3.67070403139881924395689498282, −1.07293395285700690254401514289,
0.21133174552475848237652407966, 1.90842670895994159586221310581, 2.69796174177977367386802618177, 4.18033855138143348000948265210, 5.51081090726303624404016766935, 6.89803777883300967255135324838, 7.35152949244229481110918825915, 8.214795042183409250613042069323, 9.480548372972265734968501939881, 10.06976739040495633527954117218