| L(s) = 1 | + (1.19 − 1.76i)2-s + (−1.23 + 1.21i)3-s + (−0.944 − 2.37i)4-s + (1.15 − 2.49i)5-s + (0.654 + 3.63i)6-s + (−0.389 − 1.40i)7-s + (−1.15 − 0.253i)8-s + (0.0695 − 2.99i)9-s + (−3.02 − 5.02i)10-s + (0.430 + 0.407i)11-s + (4.04 + 1.79i)12-s + (−0.374 + 3.44i)13-s + (−2.94 − 0.992i)14-s + (1.59 + 4.48i)15-s + (1.88 − 1.78i)16-s + (−3.11 − 0.866i)17-s + ⋯ |
| L(s) = 1 | + (0.846 − 1.24i)2-s + (−0.715 + 0.698i)3-s + (−0.472 − 1.18i)4-s + (0.516 − 1.11i)5-s + (0.267 + 1.48i)6-s + (−0.147 − 0.530i)7-s + (−0.407 − 0.0895i)8-s + (0.0231 − 0.999i)9-s + (−0.956 − 1.58i)10-s + (0.129 + 0.122i)11-s + (1.16 + 0.517i)12-s + (−0.103 + 0.955i)13-s + (−0.787 − 0.265i)14-s + (0.410 + 1.15i)15-s + (0.470 − 0.445i)16-s + (−0.756 − 0.210i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.196 + 0.980i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.951159 - 1.16126i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.951159 - 1.16126i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.23 - 1.21i)T \) |
| 59 | \( 1 + (7.44 + 1.88i)T \) |
| good | 2 | \( 1 + (-1.19 + 1.76i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (-1.15 + 2.49i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (0.389 + 1.40i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (-0.430 - 0.407i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.374 - 3.44i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (3.11 + 0.866i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (0.0531 + 0.0403i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-4.20 - 7.92i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (2.04 - 1.38i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (-5.98 - 7.87i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (-0.456 - 2.07i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (8.45 + 4.48i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (1.74 + 1.84i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (3.32 - 1.53i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (-4.94 + 8.21i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (5.87 + 3.98i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (2.31 - 10.5i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (-1.17 - 2.54i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (-3.15 + 9.37i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.882 + 16.2i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (2.35 - 14.3i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (-3.37 - 4.97i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (-3.60 - 10.7i)T + (-77.2 + 58.7i)T^{2} \) |
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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
−12.22419321354814871256463027134, −11.60295691157387534765544344431, −10.65702936117606704555458914438, −9.716447460545379808285964160301, −8.952796505472111173085791683761, −6.83280912528526994362657329334, −5.26741567002617779793581584593, −4.68530242032662153708852409756, −3.55123939197027081624371275445, −1.44874772816131957202400852067,
2.67413251131681920184695572666, 4.70005756761076489099471757783, 5.95924008810170369412672316015, 6.38589284932927063144369609935, 7.31431567930432932275669189299, 8.406854501961785086110813412009, 10.23495034823913787358898919910, 11.07034786911672385062944523998, 12.37353195070349438673923205836, 13.20336732523513014920939827779
