L(s) = 1 | + (1 − i)2-s + (−1 + i)3-s − 2i·4-s + (2 − i)5-s + 2i·6-s + (−2 − 2i)8-s + i·9-s + (1 − 3i)10-s + (−3 + 3i)11-s + (2 + 2i)12-s + (−3 + 3i)13-s + (−1 + 3i)15-s − 4·16-s − 4i·17-s + (1 + i)18-s + (−1 − i)19-s + ⋯ |
L(s) = 1 | + (0.707 − 0.707i)2-s + (−0.577 + 0.577i)3-s − i·4-s + (0.894 − 0.447i)5-s + 0.816i·6-s + (−0.707 − 0.707i)8-s + 0.333i·9-s + (0.316 − 0.948i)10-s + (−0.904 + 0.904i)11-s + (0.577 + 0.577i)12-s + (−0.832 + 0.832i)13-s + (−0.258 + 0.774i)15-s − 16-s − 0.970i·17-s + (0.235 + 0.235i)18-s + (−0.229 − 0.229i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.755 + 0.655i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.10789 - 0.413506i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.10789 - 0.413506i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1 + i)T \) |
| 5 | \( 1 + (-2 + i)T \) |
good | 3 | \( 1 + (1 - i)T - 3iT^{2} \) |
| 7 | \( 1 + 7T^{2} \) |
| 11 | \( 1 + (3 - 3i)T - 11iT^{2} \) |
| 13 | \( 1 + (3 - 3i)T - 13iT^{2} \) |
| 17 | \( 1 + 4iT - 17T^{2} \) |
| 19 | \( 1 + (1 + i)T + 19iT^{2} \) |
| 23 | \( 1 - 8T + 23T^{2} \) |
| 29 | \( 1 + (-3 - 3i)T + 29iT^{2} \) |
| 31 | \( 1 + 31T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 - 41T^{2} \) |
| 43 | \( 1 + (3 + 3i)T + 43iT^{2} \) |
| 47 | \( 1 + 2iT - 47T^{2} \) |
| 53 | \( 1 + (-9 - 9i)T + 53iT^{2} \) |
| 59 | \( 1 + (-9 + 9i)T - 59iT^{2} \) |
| 61 | \( 1 + (5 + 5i)T + 61iT^{2} \) |
| 67 | \( 1 + (-3 + 3i)T - 67iT^{2} \) |
| 71 | \( 1 + 6iT - 71T^{2} \) |
| 73 | \( 1 + 6T + 73T^{2} \) |
| 79 | \( 1 - 8T + 79T^{2} \) |
| 83 | \( 1 + (9 - 9i)T - 83iT^{2} \) |
| 89 | \( 1 - 12iT - 89T^{2} \) |
| 97 | \( 1 - 12iT - 97T^{2} \) |
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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
−14.04901381589848517669287105152, −13.12580103815786168627567314299, −12.17801621362332164398493210095, −10.96056544901509384935604857787, −10.07059388567639789516193372782, −9.228785443790332110863380008958, −6.91565115638612820206505696023, −5.16957075528633276263265883990, −4.81303490109649790741175582599, −2.35800388573391443252054775612,
3.00055124653600007170672131703, 5.28560249338780965286128836829, 6.13044312346254729524365106867, 7.18837648543051429883700073371, 8.550786391460375997190461702687, 10.26075897927273957609018588390, 11.49524970232935581612553876823, 12.85605969118348066594036764241, 13.21263327872961666426845699339, 14.57742198816264828588736620394