| L(s) = 1 | + (−1.28 − 0.850i)2-s + (0.151 + 0.354i)4-s + (1.87 − 2.58i)5-s + (−0.371 + 0.990i)7-s + (−0.445 + 2.45i)8-s + (−4.61 + 1.73i)10-s + (0.641 − 4.73i)11-s + (1.50 − 0.204i)13-s + (1.32 − 0.960i)14-s + (3.19 − 3.34i)16-s + (1.00 − 3.10i)17-s + (−4.90 + 2.92i)19-s + (1.19 + 0.273i)20-s + (−4.85 + 5.55i)22-s + (3.99 − 5.01i)23-s + ⋯ |
| L(s) = 1 | + (−0.911 − 0.601i)2-s + (0.0757 + 0.177i)4-s + (0.838 − 1.15i)5-s + (−0.140 + 0.374i)7-s + (−0.157 + 0.867i)8-s + (−1.45 + 0.547i)10-s + (0.193 − 1.42i)11-s + (0.418 − 0.0567i)13-s + (0.353 − 0.256i)14-s + (0.798 − 0.835i)16-s + (0.244 − 0.753i)17-s + (−1.12 + 0.671i)19-s + (0.267 + 0.0611i)20-s + (−1.03 + 1.18i)22-s + (0.833 − 1.04i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 639 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.857 + 0.515i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.236573 - 0.853021i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.236573 - 0.853021i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 71 | \( 1 + (6.70 + 5.09i)T \) |
| good | 2 | \( 1 + (1.28 + 0.850i)T + (0.786 + 1.83i)T^{2} \) |
| 5 | \( 1 + (-1.87 + 2.58i)T + (-1.54 - 4.75i)T^{2} \) |
| 7 | \( 1 + (0.371 - 0.990i)T + (-5.27 - 4.60i)T^{2} \) |
| 11 | \( 1 + (-0.641 + 4.73i)T + (-10.6 - 2.92i)T^{2} \) |
| 13 | \( 1 + (-1.50 + 0.204i)T + (12.5 - 3.45i)T^{2} \) |
| 17 | \( 1 + (-1.00 + 3.10i)T + (-13.7 - 9.99i)T^{2} \) |
| 19 | \( 1 + (4.90 - 2.92i)T + (9.00 - 16.7i)T^{2} \) |
| 23 | \( 1 + (-3.99 + 5.01i)T + (-5.11 - 22.4i)T^{2} \) |
| 29 | \( 1 + (-1.87 - 2.15i)T + (-3.89 + 28.7i)T^{2} \) |
| 31 | \( 1 + (-0.432 + 0.413i)T + (1.39 - 30.9i)T^{2} \) |
| 37 | \( 1 + (-0.542 - 0.679i)T + (-8.23 + 36.0i)T^{2} \) |
| 41 | \( 1 + (8.80 - 4.24i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (0.369 + 8.23i)T + (-42.8 + 3.85i)T^{2} \) |
| 47 | \( 1 + (11.3 + 1.02i)T + (46.2 + 8.39i)T^{2} \) |
| 53 | \( 1 + (-1.38 + 3.24i)T + (-36.6 - 38.3i)T^{2} \) |
| 59 | \( 1 + (2.65 + 4.93i)T + (-32.5 + 49.2i)T^{2} \) |
| 61 | \( 1 + (-0.00732 - 0.0195i)T + (-45.9 + 40.1i)T^{2} \) |
| 67 | \( 1 + (1.39 - 0.595i)T + (46.3 - 48.4i)T^{2} \) |
| 73 | \( 1 + (-4.03 + 6.11i)T + (-28.6 - 67.1i)T^{2} \) |
| 79 | \( 1 + (-4.58 - 0.832i)T + (73.9 + 27.7i)T^{2} \) |
| 83 | \( 1 + (14.1 - 7.59i)T + (45.7 - 69.2i)T^{2} \) |
| 89 | \( 1 + (1.81 + 0.774i)T + (61.5 + 64.3i)T^{2} \) |
| 97 | \( 1 + (0.197 - 0.409i)T + (-60.4 - 75.8i)T^{2} \) |
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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
−10.13567702094923098614095597326, −9.260312964761943496430331428201, −8.620888278312644867830550076371, −8.297927266498844697562341391566, −6.46413184704649162487568461329, −5.63170755708971762694468752741, −4.83446064083493526673516525921, −3.12086121870046117349848760543, −1.80273257603116725561365845828, −0.67320150419566838251278484204,
1.74436890810619488951244445707, 3.19475445589574173847889837337, 4.41295272808973714358485414916, 5.98945281830215957934169815075, 6.81863566972655915156845859734, 7.19302617304351462653950644955, 8.312837659771616630524614498711, 9.266377125922792789880570439485, 10.04238475011924223712021082282, 10.44567558668313305269090349705
