| L(s) = 1 | + (−0.915 + 2.09i)2-s + (1.69 + 0.332i)3-s + (−2.20 − 2.37i)4-s + (−1.59 − 0.240i)5-s + (−2.25 + 3.26i)6-s + (2.61 + 2.43i)7-s + (2.67 − 0.934i)8-s + (2.77 + 1.12i)9-s + (1.96 − 3.12i)10-s + (−2.89 + 2.49i)11-s + (−2.95 − 4.76i)12-s + (−2.15 + 3.15i)13-s + (−7.49 + 3.27i)14-s + (−2.63 − 0.938i)15-s + (−0.000114 + 0.00152i)16-s + (−2.40 − 2.40i)17-s + ⋯ |
| L(s) = 1 | + (−0.647 + 1.48i)2-s + (0.981 + 0.191i)3-s + (−1.10 − 1.18i)4-s + (−0.713 − 0.107i)5-s + (−0.919 + 1.33i)6-s + (0.990 + 0.918i)7-s + (0.944 − 0.330i)8-s + (0.926 + 0.376i)9-s + (0.620 − 0.988i)10-s + (−0.873 + 0.751i)11-s + (−0.852 − 1.37i)12-s + (−0.596 + 0.874i)13-s + (−2.00 + 0.874i)14-s + (−0.679 − 0.242i)15-s + (−2.86e−5 + 0.000382i)16-s + (−0.582 − 0.582i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 261 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.941 - 0.335i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.180703 + 1.04545i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.180703 + 1.04545i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.69 - 0.332i)T \) |
| 29 | \( 1 + (-5.35 + 0.564i)T \) |
| good | 2 | \( 1 + (0.915 - 2.09i)T + (-1.36 - 1.46i)T^{2} \) |
| 5 | \( 1 + (1.59 + 0.240i)T + (4.77 + 1.47i)T^{2} \) |
| 7 | \( 1 + (-2.61 - 2.43i)T + (0.523 + 6.98i)T^{2} \) |
| 11 | \( 1 + (2.89 - 2.49i)T + (1.63 - 10.8i)T^{2} \) |
| 13 | \( 1 + (2.15 - 3.15i)T + (-4.74 - 12.1i)T^{2} \) |
| 17 | \( 1 + (2.40 + 2.40i)T + 17iT^{2} \) |
| 19 | \( 1 + (-3.94 - 2.48i)T + (8.24 + 17.1i)T^{2} \) |
| 23 | \( 1 + (-1.21 - 0.476i)T + (16.8 + 15.6i)T^{2} \) |
| 31 | \( 1 + (-5.64 + 4.16i)T + (9.13 - 29.6i)T^{2} \) |
| 37 | \( 1 + (3.43 + 9.83i)T + (-28.9 + 23.0i)T^{2} \) |
| 41 | \( 1 + (-3.98 - 1.06i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (4.24 - 5.74i)T + (-12.6 - 41.0i)T^{2} \) |
| 47 | \( 1 + (-1.51 - 1.75i)T + (-7.00 + 46.4i)T^{2} \) |
| 53 | \( 1 + (1.15 + 0.918i)T + (11.7 + 51.6i)T^{2} \) |
| 59 | \( 1 + (-10.0 + 5.78i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.11 + 0.0417i)T + (60.8 - 4.55i)T^{2} \) |
| 67 | \( 1 + (-1.39 + 0.104i)T + (66.2 - 9.98i)T^{2} \) |
| 71 | \( 1 + (-11.9 + 5.73i)T + (44.2 - 55.5i)T^{2} \) |
| 73 | \( 1 + (-0.714 + 6.34i)T + (-71.1 - 16.2i)T^{2} \) |
| 79 | \( 1 + (1.64 - 8.71i)T + (-73.5 - 28.8i)T^{2} \) |
| 83 | \( 1 + (2.03 - 6.58i)T + (-68.5 - 46.7i)T^{2} \) |
| 89 | \( 1 + (2.47 - 0.278i)T + (86.7 - 19.8i)T^{2} \) |
| 97 | \( 1 + (2.31 + 4.37i)T + (-54.6 + 80.1i)T^{2} \) |
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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
−12.41931326477995828274448505565, −11.48114569589194609188336496226, −9.885579523328367468806188743685, −9.216621422186013359213762973662, −8.193587865288252902465918680726, −7.81007147146097322120419369825, −6.90577783008108857065722657180, −5.26812092792742769304634000730, −4.47175316428764709032433805162, −2.37474845968220503583118293725,
0.990433812661724911992285190107, 2.63549049279911243874134771698, 3.56386763142160415042657052175, 4.76025465235243313621686762614, 7.16865260585622131796772104936, 8.188365186900421444969140475894, 8.480317680630385396017177195442, 9.977827618791021443236614605626, 10.53854902177679826971121091192, 11.40743732405211349394220668890
