L(s) = 1 | + (−0.5 + 0.866i)2-s + (−1 − 1.73i)3-s + (0.500 + 0.866i)4-s + (1 − 1.73i)5-s + 1.99·6-s − 3·8-s + (−0.499 + 0.866i)9-s + (0.999 + 1.73i)10-s + (−0.5 − 0.866i)11-s + (0.999 − 1.73i)12-s + 4·13-s − 3.99·15-s + (0.500 − 0.866i)16-s + (−2 − 3.46i)17-s + (−0.499 − 0.866i)18-s + ⋯ |
L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.577 − 0.999i)3-s + (0.250 + 0.433i)4-s + (0.447 − 0.774i)5-s + 0.816·6-s − 1.06·8-s + (−0.166 + 0.288i)9-s + (0.316 + 0.547i)10-s + (−0.150 − 0.261i)11-s + (0.288 − 0.499i)12-s + 1.10·13-s − 1.03·15-s + (0.125 − 0.216i)16-s + (−0.485 − 0.840i)17-s + (−0.117 − 0.204i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 539 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 + 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.834465 - 0.555071i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.834465 - 0.555071i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 + (0.5 + 0.866i)T \) |
good | 2 | \( 1 + (0.5 - 0.866i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (1 + 1.73i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-1 + 1.73i)T + (-2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 - 4T + 13T^{2} \) |
| 17 | \( 1 + (2 + 3.46i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-2 + 3.46i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + 6T + 29T^{2} \) |
| 31 | \( 1 + (5 + 8.66i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3 + 5.19i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 4T + 41T^{2} \) |
| 43 | \( 1 - 12T + 43T^{2} \) |
| 47 | \( 1 + (-5 + 8.66i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-3 - 5.19i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (1 + 1.73i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4 + 6.92i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (-4 - 6.92i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (4 - 6.92i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 83T^{2} \) |
| 89 | \( 1 + (-3 + 5.19i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + 10T + 97T^{2} \) |
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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
−10.97648437404363202733622829791, −9.255607369495426077510239146831, −8.915068724933190951973655638418, −7.71430761165178166621451971517, −7.14289908071580893222445482721, −6.07904090595396138305132622536, −5.61367086313176459278527558449, −3.95104844302004535037014515030, −2.29747422209458246179664832279, −0.71514091490976460271343380200,
1.65851478706095746306995211881, 3.04092962074346211991971874253, 4.21974800368305056021870637145, 5.58613287538501873009718315485, 6.12008228585343471143273714559, 7.24874020720812793931931323741, 8.787498650057598666525220333178, 9.543965587989615812576794801884, 10.36149645196693557360741551364, 10.94631785250250540258437311197