| L(s) = 1 | + (−0.841 + 0.540i)2-s + (−0.142 + 0.989i)3-s + (0.415 − 0.909i)4-s + (2.55 − 0.750i)5-s + (−0.415 − 0.909i)6-s + (0.654 + 0.755i)7-s + (0.142 + 0.989i)8-s + (−0.959 − 0.281i)9-s + (−1.74 + 2.01i)10-s + (−1.70 − 1.09i)11-s + (0.841 + 0.540i)12-s + (−2.79 + 3.23i)13-s + (−0.959 − 0.281i)14-s + (0.379 + 2.63i)15-s + (−0.654 − 0.755i)16-s + (1.08 + 2.37i)17-s + ⋯ |
| L(s) = 1 | + (−0.594 + 0.382i)2-s + (−0.0821 + 0.571i)3-s + (0.207 − 0.454i)4-s + (1.14 − 0.335i)5-s + (−0.169 − 0.371i)6-s + (0.247 + 0.285i)7-s + (0.0503 + 0.349i)8-s + (−0.319 − 0.0939i)9-s + (−0.551 + 0.637i)10-s + (−0.514 − 0.330i)11-s + (0.242 + 0.156i)12-s + (−0.776 + 0.895i)13-s + (−0.256 − 0.0752i)14-s + (0.0979 + 0.681i)15-s + (−0.163 − 0.188i)16-s + (0.263 + 0.576i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.269 - 0.962i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.269 - 0.962i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.741498 + 0.977613i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.741498 + 0.977613i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.841 - 0.540i)T \) |
| 3 | \( 1 + (0.142 - 0.989i)T \) |
| 7 | \( 1 + (-0.654 - 0.755i)T \) |
| 23 | \( 1 + (1.22 - 4.63i)T \) |
| good | 5 | \( 1 + (-2.55 + 0.750i)T + (4.20 - 2.70i)T^{2} \) |
| 11 | \( 1 + (1.70 + 1.09i)T + (4.56 + 10.0i)T^{2} \) |
| 13 | \( 1 + (2.79 - 3.23i)T + (-1.85 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-1.08 - 2.37i)T + (-11.1 + 12.8i)T^{2} \) |
| 19 | \( 1 + (0.679 - 1.48i)T + (-12.4 - 14.3i)T^{2} \) |
| 29 | \( 1 + (-3.87 - 8.49i)T + (-18.9 + 21.9i)T^{2} \) |
| 31 | \( 1 + (0.105 + 0.735i)T + (-29.7 + 8.73i)T^{2} \) |
| 37 | \( 1 + (-6.50 - 1.91i)T + (31.1 + 20.0i)T^{2} \) |
| 41 | \( 1 + (0.774 - 0.227i)T + (34.4 - 22.1i)T^{2} \) |
| 43 | \( 1 + (-1.09 + 7.58i)T + (-41.2 - 12.1i)T^{2} \) |
| 47 | \( 1 - 12.7T + 47T^{2} \) |
| 53 | \( 1 + (-7.66 - 8.84i)T + (-7.54 + 52.4i)T^{2} \) |
| 59 | \( 1 + (0.759 - 0.876i)T + (-8.39 - 58.3i)T^{2} \) |
| 61 | \( 1 + (0.439 + 3.05i)T + (-58.5 + 17.1i)T^{2} \) |
| 67 | \( 1 + (6.29 - 4.04i)T + (27.8 - 60.9i)T^{2} \) |
| 71 | \( 1 + (3.05 - 1.96i)T + (29.4 - 64.5i)T^{2} \) |
| 73 | \( 1 + (-2.26 + 4.95i)T + (-47.8 - 55.1i)T^{2} \) |
| 79 | \( 1 + (4.61 - 5.32i)T + (-11.2 - 78.1i)T^{2} \) |
| 83 | \( 1 + (2.52 + 0.740i)T + (69.8 + 44.8i)T^{2} \) |
| 89 | \( 1 + (1.10 - 7.69i)T + (-85.3 - 25.0i)T^{2} \) |
| 97 | \( 1 + (10.9 - 3.21i)T + (81.6 - 52.4i)T^{2} \) |
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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
−10.17956416125347257453130332254, −9.303431599778370340162340271481, −8.892036944879032729734878673585, −7.87836467488476046168951449677, −6.87001239834857012755314381368, −5.74726909517978754034033843708, −5.40120303263095625307238638358, −4.23646168510919637668292351164, −2.63936718741269588124695959875, −1.52004830646019874667345788300,
0.70739593425000706031135955691, 2.27355615397563067424657774118, 2.71970366994671741837992835770, 4.49385587399430371351271025221, 5.60744590538081504506019893424, 6.44257661414306116938039431465, 7.42703183391426795265115992829, 7.994409155842564643608685705063, 9.059360035024839886489114356596, 10.12612622619636616216191207336
