L(s) = 1 | + (−0.142 − 0.989i)2-s + (0.415 − 0.909i)3-s + (−0.959 + 0.281i)4-s + (−2.42 + 2.80i)5-s + (−0.959 − 0.281i)6-s + (−0.841 + 0.540i)7-s + (0.415 + 0.909i)8-s + (−0.654 − 0.755i)9-s + (3.11 + 2.00i)10-s + (0.504 − 3.51i)11-s + (−0.142 + 0.989i)12-s + (−1.29 − 0.832i)13-s + (0.654 + 0.755i)14-s + (1.54 + 3.37i)15-s + (0.841 − 0.540i)16-s + (4.65 + 1.36i)17-s + ⋯ |
L(s) = 1 | + (−0.100 − 0.699i)2-s + (0.239 − 0.525i)3-s + (−0.479 + 0.140i)4-s + (−1.08 + 1.25i)5-s + (−0.391 − 0.115i)6-s + (−0.317 + 0.204i)7-s + (0.146 + 0.321i)8-s + (−0.218 − 0.251i)9-s + (0.986 + 0.633i)10-s + (0.152 − 1.05i)11-s + (−0.0410 + 0.285i)12-s + (−0.359 − 0.230i)13-s + (0.175 + 0.201i)14-s + (0.397 + 0.870i)15-s + (0.210 − 0.135i)16-s + (1.12 + 0.331i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 966 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.294 + 0.955i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.930620 - 0.686753i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.930620 - 0.686753i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.142 + 0.989i)T \) |
| 3 | \( 1 + (-0.415 + 0.909i)T \) |
| 7 | \( 1 + (0.841 - 0.540i)T \) |
| 23 | \( 1 + (-4.70 + 0.921i)T \) |
good | 5 | \( 1 + (2.42 - 2.80i)T + (-0.711 - 4.94i)T^{2} \) |
| 11 | \( 1 + (-0.504 + 3.51i)T + (-10.5 - 3.09i)T^{2} \) |
| 13 | \( 1 + (1.29 + 0.832i)T + (5.40 + 11.8i)T^{2} \) |
| 17 | \( 1 + (-4.65 - 1.36i)T + (14.3 + 9.19i)T^{2} \) |
| 19 | \( 1 + (0.0720 - 0.0211i)T + (15.9 - 10.2i)T^{2} \) |
| 29 | \( 1 + (-9.19 - 2.69i)T + (24.3 + 15.6i)T^{2} \) |
| 31 | \( 1 + (3.19 + 7.00i)T + (-20.3 + 23.4i)T^{2} \) |
| 37 | \( 1 + (-6.22 - 7.18i)T + (-5.26 + 36.6i)T^{2} \) |
| 41 | \( 1 + (-4.33 + 5.00i)T + (-5.83 - 40.5i)T^{2} \) |
| 43 | \( 1 + (-4.25 + 9.32i)T + (-28.1 - 32.4i)T^{2} \) |
| 47 | \( 1 + 0.339T + 47T^{2} \) |
| 53 | \( 1 + (-8.01 + 5.15i)T + (22.0 - 48.2i)T^{2} \) |
| 59 | \( 1 + (6.97 + 4.48i)T + (24.5 + 53.6i)T^{2} \) |
| 61 | \( 1 + (-2.69 - 5.89i)T + (-39.9 + 46.1i)T^{2} \) |
| 67 | \( 1 + (0.191 + 1.33i)T + (-64.2 + 18.8i)T^{2} \) |
| 71 | \( 1 + (-0.745 - 5.18i)T + (-68.1 + 20.0i)T^{2} \) |
| 73 | \( 1 + (12.5 - 3.67i)T + (61.4 - 39.4i)T^{2} \) |
| 79 | \( 1 + (6.16 + 3.96i)T + (32.8 + 71.8i)T^{2} \) |
| 83 | \( 1 + (-3.10 - 3.58i)T + (-11.8 + 82.1i)T^{2} \) |
| 89 | \( 1 + (-0.675 + 1.47i)T + (-58.2 - 67.2i)T^{2} \) |
| 97 | \( 1 + (6.85 - 7.91i)T + (-13.8 - 96.0i)T^{2} \) |
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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
−10.11144445647314103667654385519, −8.940855613191293738848268229255, −8.154980088407333548676061183427, −7.45420028959535075857596920790, −6.60621961043583429263836264907, −5.60125075296349758785683109736, −4.06138549727995121043194555319, −3.19023853179982762937514324734, −2.65362576489869340990798750927, −0.74158536057344276221928488888,
1.00825338192090984139720175693, 3.11848914648680072171567872652, 4.44590688115748988853381352372, 4.60705480061117699351603546613, 5.74136422944375152593923476094, 7.12114701112621568732134187050, 7.67155373436972228495801336738, 8.495280735895427742163740949778, 9.303328625233518264654465147457, 9.794504815461725626876748307002