| L(s) = 1 | + (−0.460 + 0.679i)2-s + (1.70 − 0.281i)3-s + (0.490 + 1.23i)4-s + (0.472 − 1.02i)5-s + (−0.596 + 1.29i)6-s + (0.500 + 1.80i)7-s + (−2.66 − 0.587i)8-s + (2.84 − 0.962i)9-s + (0.475 + 0.791i)10-s + (−1.67 − 1.58i)11-s + (1.18 + 1.96i)12-s + (−0.00701 + 0.0645i)13-s + (−1.45 − 0.490i)14-s + (0.519 − 1.87i)15-s + (−0.297 + 0.281i)16-s + (1.12 + 0.311i)17-s + ⋯ |
| L(s) = 1 | + (−0.325 + 0.480i)2-s + (0.986 − 0.162i)3-s + (0.245 + 0.615i)4-s + (0.211 − 0.456i)5-s + (−0.243 + 0.527i)6-s + (0.189 + 0.681i)7-s + (−0.942 − 0.207i)8-s + (0.947 − 0.320i)9-s + (0.150 + 0.250i)10-s + (−0.503 − 0.477i)11-s + (0.342 + 0.567i)12-s + (−0.00194 + 0.0178i)13-s + (−0.389 − 0.131i)14-s + (0.134 − 0.484i)15-s + (−0.0743 + 0.0704i)16-s + (0.272 + 0.0755i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.686 - 0.727i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.28215 + 0.552945i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.28215 + 0.552945i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.70 + 0.281i)T \) |
| 59 | \( 1 + (7.53 + 1.49i)T \) |
| good | 2 | \( 1 + (0.460 - 0.679i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (-0.472 + 1.02i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (-0.500 - 1.80i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (1.67 + 1.58i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.00701 - 0.0645i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (-1.12 - 0.311i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (2.84 + 2.16i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (-0.0691 - 0.130i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (4.18 - 2.83i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (1.05 + 1.38i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (1.76 + 8.01i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (3.40 + 1.80i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (7.42 + 7.83i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (-5.61 + 2.59i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (4.90 - 8.16i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (-5.16 - 3.49i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (-0.0126 + 0.0576i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (-0.791 - 1.71i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (-3.52 + 10.4i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (0.795 - 14.6i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (1.95 - 11.9i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (-9.17 - 13.5i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (-4.38 - 12.9i)T + (-77.2 + 58.7i)T^{2} \) |
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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
−12.83572434687822860453276507946, −12.15234395524866527838462201478, −10.77627417591798046928844916507, −9.214900523018988772081375079437, −8.760077172788980940340142115082, −7.85603731127425066028670011011, −6.86304705280323020951873522631, −5.42615131152859742532608943801, −3.63598109459877702845613506318, −2.31498977952651006198542840373,
1.81939184123744743475356353620, 3.12729088789581399935622492603, 4.72193521909501965727035053831, 6.38853068777204677544275569979, 7.53342619213925753210835399984, 8.646338174151001354791214358537, 9.947106751349591867082972212091, 10.23580546356773477644122118810, 11.27421458931417251811052943834, 12.62248825482966872459387198597
