L(s) = 1 | − 1.85·2-s + (0.240 + 0.415i)3-s + 1.42·4-s + (0.854 − 1.48i)5-s + (−0.444 − 0.770i)6-s + (−0.531 − 0.921i)7-s + 1.05·8-s + (1.38 − 2.39i)9-s + (−1.58 + 2.74i)10-s + (−1.35 + 2.34i)11-s + (0.343 + 0.594i)12-s + (−0.5 + 0.866i)13-s + (0.984 + 1.70i)14-s + 0.821·15-s − 4.81·16-s + (−2.30 − 3.98i)17-s + ⋯ |
L(s) = 1 | − 1.30·2-s + (0.138 + 0.240i)3-s + 0.714·4-s + (0.382 − 0.662i)5-s + (−0.181 − 0.314i)6-s + (−0.201 − 0.348i)7-s + 0.374·8-s + (0.461 − 0.799i)9-s + (−0.500 + 0.867i)10-s + (−0.407 + 0.706i)11-s + (0.0990 + 0.171i)12-s + (−0.138 + 0.240i)13-s + (0.263 + 0.455i)14-s + 0.212·15-s − 1.20·16-s + (−0.558 − 0.966i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0999 + 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0999 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.468633 - 0.423922i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.468633 - 0.423922i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 + (0.5 - 0.866i)T \) |
| 31 | \( 1 + (3.77 + 4.08i)T \) |
good | 2 | \( 1 + 1.85T + 2T^{2} \) |
| 3 | \( 1 + (-0.240 - 0.415i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.854 + 1.48i)T + (-2.5 - 4.33i)T^{2} \) |
| 7 | \( 1 + (0.531 + 0.921i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (1.35 - 2.34i)T + (-5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (2.30 + 3.98i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.35 + 4.07i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 0.635T + 23T^{2} \) |
| 29 | \( 1 - 8.27T + 29T^{2} \) |
| 37 | \( 1 + (5.38 + 9.32i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-2.87 + 4.97i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (3.63 + 6.30i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 1.75T + 47T^{2} \) |
| 53 | \( 1 + (0.713 - 1.23i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.20 + 2.08i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 - 11.0T + 61T^{2} \) |
| 67 | \( 1 + (2.39 - 4.15i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.06 - 10.4i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (3.91 - 6.77i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.09 + 5.36i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.24 + 7.35i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 - 7.00T + 89T^{2} \) |
| 97 | \( 1 + 9.12T + 97T^{2} \) |
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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
−10.60384824913286097080184702826, −9.958060383438939459079643099595, −9.075004586977919313669547271363, −8.814483742303696767194539841719, −7.31930775338651493451342331214, −6.84217162253040557710195510353, −5.10680956941729110604160948354, −4.15263684930393412888274006949, −2.23571966893852891273220142023, −0.64270724016019112699910672773,
1.64114733283905714613143450961, 2.86645559073583732776438670594, 4.66391732899853111723704163119, 6.14035642840067394785617402529, 6.97828621008759611469413328111, 8.206342786220155916767543979783, 8.443495954338225241824518361350, 9.777691228144992927440569491613, 10.52403904774791521721561145897, 10.83942034950444764838937913483