| L(s) = 1 | + (0.230 − 1.60i)2-s + (0.928 + 2.03i)3-s + (−0.592 − 0.174i)4-s + (2.11 + 2.44i)5-s + (3.47 − 1.01i)6-s + (−1.03 − 0.668i)7-s + (0.928 − 2.03i)8-s + (−1.30 + 1.51i)9-s + (4.40 − 2.83i)10-s + (0.108 + 0.756i)11-s + (−0.196 − 1.36i)12-s + (2.52 − 1.62i)13-s + (−1.30 + 1.51i)14-s + (−3.00 + 6.58i)15-s + (−4.08 − 2.62i)16-s + (−5.02 + 1.47i)17-s + ⋯ |
| L(s) = 1 | + (0.162 − 1.13i)2-s + (0.536 + 1.17i)3-s + (−0.296 − 0.0870i)4-s + (0.947 + 1.09i)5-s + (1.41 − 0.416i)6-s + (−0.393 − 0.252i)7-s + (0.328 − 0.719i)8-s + (−0.436 + 0.503i)9-s + (1.39 − 0.895i)10-s + (0.0327 + 0.227i)11-s + (−0.0567 − 0.394i)12-s + (0.699 − 0.449i)13-s + (−0.350 + 0.403i)14-s + (−0.776 + 1.69i)15-s + (−1.02 − 0.656i)16-s + (−1.21 + 0.357i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0235i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.27254 + 0.0267637i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.27254 + 0.0267637i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 23 | \( 1 \) |
| good | 2 | \( 1 + (-0.230 + 1.60i)T + (-1.91 - 0.563i)T^{2} \) |
| 3 | \( 1 + (-0.928 - 2.03i)T + (-1.96 + 2.26i)T^{2} \) |
| 5 | \( 1 + (-2.11 - 2.44i)T + (-0.711 + 4.94i)T^{2} \) |
| 7 | \( 1 + (1.03 + 0.668i)T + (2.90 + 6.36i)T^{2} \) |
| 11 | \( 1 + (-0.108 - 0.756i)T + (-10.5 + 3.09i)T^{2} \) |
| 13 | \( 1 + (-2.52 + 1.62i)T + (5.40 - 11.8i)T^{2} \) |
| 17 | \( 1 + (5.02 - 1.47i)T + (14.3 - 9.19i)T^{2} \) |
| 19 | \( 1 + (-1.91 - 0.563i)T + (15.9 + 10.2i)T^{2} \) |
| 29 | \( 1 + (-2.87 + 0.845i)T + (24.3 - 15.6i)T^{2} \) |
| 31 | \( 1 + (2.78 - 6.10i)T + (-20.3 - 23.4i)T^{2} \) |
| 37 | \( 1 + (2.11 - 2.44i)T + (-5.26 - 36.6i)T^{2} \) |
| 41 | \( 1 + (3.58 + 4.13i)T + (-5.83 + 40.5i)T^{2} \) |
| 43 | \( 1 + (-28.1 + 32.4i)T^{2} \) |
| 47 | \( 1 - 2.23T + 47T^{2} \) |
| 53 | \( 1 + (7.12 + 4.58i)T + (22.0 + 48.2i)T^{2} \) |
| 59 | \( 1 + (2.07 - 1.33i)T + (24.5 - 53.6i)T^{2} \) |
| 61 | \( 1 + (-4.54 + 9.95i)T + (-39.9 - 46.1i)T^{2} \) |
| 67 | \( 1 + (-1.02 + 7.16i)T + (-64.2 - 18.8i)T^{2} \) |
| 71 | \( 1 + (1.10 - 7.68i)T + (-68.1 - 20.0i)T^{2} \) |
| 73 | \( 1 + (14.8 + 4.35i)T + (61.4 + 39.4i)T^{2} \) |
| 79 | \( 1 + (-5.84 + 3.75i)T + (32.8 - 71.8i)T^{2} \) |
| 83 | \( 1 + (-8.66 + 10.0i)T + (-11.8 - 82.1i)T^{2} \) |
| 89 | \( 1 + (0.634 + 1.38i)T + (-58.2 + 67.2i)T^{2} \) |
| 97 | \( 1 + (2.81 + 3.24i)T + (-13.8 + 96.0i)T^{2} \) |
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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
−10.59194089381954762973927699542, −10.23095711684748141662241876635, −9.558210677090221137013699916050, −8.651632413971439938023385616186, −7.04044022520330944599968697150, −6.29549287457502237163071162446, −4.79683331839251770143443680456, −3.60053195531900179911322072823, −3.09860649485615642024116210163, −1.93281287886989059711812707413,
1.45839984849032635853241413954, 2.50029619237878292672135820700, 4.54180239726472881492503551454, 5.63503355669127228861971760866, 6.35631059377617218433838718434, 7.07215084041974883298894652941, 8.101368987902026345289391992490, 8.804842877533344780065859548822, 9.420225566430021816451645700766, 10.92590862403668904418903585729
