L(s) = 1 | + (0.279 − 0.0749i)3-s + (0.774 − 2.09i)5-s + (−2.64 − 0.126i)7-s + (−2.52 + 1.45i)9-s + (2.81 − 4.87i)11-s + (1.42 + 1.42i)13-s + (0.0593 − 0.645i)15-s + (−1.37 − 5.12i)17-s + (−1.94 − 3.37i)19-s + (−0.749 + 0.162i)21-s + (−1.08 − 0.290i)23-s + (−3.80 − 3.24i)25-s + (−1.21 + 1.21i)27-s − 3.15i·29-s + (3.33 + 1.92i)31-s + ⋯ |
L(s) = 1 | + (0.161 − 0.0432i)3-s + (0.346 − 0.938i)5-s + (−0.998 − 0.0477i)7-s + (−0.841 + 0.486i)9-s + (0.848 − 1.46i)11-s + (0.396 + 0.396i)13-s + (0.0153 − 0.166i)15-s + (−0.333 − 1.24i)17-s + (−0.446 − 0.773i)19-s + (−0.163 + 0.0355i)21-s + (−0.226 − 0.0606i)23-s + (−0.760 − 0.649i)25-s + (−0.233 + 0.233i)27-s − 0.585i·29-s + (0.598 + 0.345i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.242 + 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.723845 - 0.926715i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.723845 - 0.926715i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-0.774 + 2.09i)T \) |
| 7 | \( 1 + (2.64 + 0.126i)T \) |
good | 3 | \( 1 + (-0.279 + 0.0749i)T + (2.59 - 1.5i)T^{2} \) |
| 11 | \( 1 + (-2.81 + 4.87i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.42 - 1.42i)T + 13iT^{2} \) |
| 17 | \( 1 + (1.37 + 5.12i)T + (-14.7 + 8.5i)T^{2} \) |
| 19 | \( 1 + (1.94 + 3.37i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.08 + 0.290i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + 3.15iT - 29T^{2} \) |
| 31 | \( 1 + (-3.33 - 1.92i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (1.30 - 4.86i)T + (-32.0 - 18.5i)T^{2} \) |
| 41 | \( 1 + 7.21iT - 41T^{2} \) |
| 43 | \( 1 + (1.85 - 1.85i)T - 43iT^{2} \) |
| 47 | \( 1 + (-5.69 - 1.52i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (0.357 + 1.33i)T + (-45.8 + 26.5i)T^{2} \) |
| 59 | \( 1 + (-2.73 + 4.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.99 - 2.30i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.816 + 0.218i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 4.77T + 71T^{2} \) |
| 73 | \( 1 + (-5.42 + 1.45i)T + (63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-5.41 + 3.12i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-5.67 - 5.67i)T + 83iT^{2} \) |
| 89 | \( 1 + (-5.96 - 10.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-6.63 + 6.63i)T - 97iT^{2} \) |
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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
−10.51930613221374777972382791651, −9.215894047608027071218632930039, −9.002956410530195454303937028129, −8.100604863579955806450665834513, −6.65700403802769976265080983605, −5.98555602510783672537133268584, −4.94754731306372885170635919944, −3.67249529245420932854785893027, −2.52121145505864454395075735572, −0.64548591966137465853417544577,
2.01084394670280486148939621369, 3.26438666517152995040300980615, 4.09674365997693732403006813462, 5.89776760966281347446457535657, 6.36993710242119792670400876806, 7.25831114014897542123533368591, 8.478618704836369419428839681812, 9.415978183890105524197555150479, 10.09380756185546739683771779290, 10.83067792311240872057603662491