| L(s) = 1 | + (0.154 − 0.227i)2-s + (0.979 + 1.42i)3-s + (0.712 + 1.78i)4-s + (1.09 − 2.37i)5-s + (0.476 − 0.00256i)6-s + (−0.787 − 2.83i)7-s + (1.05 + 0.232i)8-s + (−1.08 + 2.79i)9-s + (−0.370 − 0.616i)10-s + (1.07 + 1.02i)11-s + (−1.85 + 2.76i)12-s + (−0.324 + 2.98i)13-s + (−0.767 − 0.258i)14-s + (4.46 − 0.756i)15-s + (−2.57 + 2.44i)16-s + (−6.60 − 1.83i)17-s + ⋯ |
| L(s) = 1 | + (0.109 − 0.161i)2-s + (0.565 + 0.824i)3-s + (0.356 + 0.893i)4-s + (0.490 − 1.06i)5-s + (0.194 − 0.00104i)6-s + (−0.297 − 1.07i)7-s + (0.372 + 0.0820i)8-s + (−0.360 + 0.932i)9-s + (−0.117 − 0.194i)10-s + (0.325 + 0.308i)11-s + (−0.535 + 0.799i)12-s + (−0.0899 + 0.826i)13-s + (−0.205 − 0.0691i)14-s + (1.15 − 0.195i)15-s + (−0.644 + 0.610i)16-s + (−1.60 − 0.444i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.52248 + 0.350429i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.52248 + 0.350429i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-0.979 - 1.42i)T \) |
| 59 | \( 1 + (6.44 - 4.17i)T \) |
| good | 2 | \( 1 + (-0.154 + 0.227i)T + (-0.740 - 1.85i)T^{2} \) |
| 5 | \( 1 + (-1.09 + 2.37i)T + (-3.23 - 3.81i)T^{2} \) |
| 7 | \( 1 + (0.787 + 2.83i)T + (-5.99 + 3.60i)T^{2} \) |
| 11 | \( 1 + (-1.07 - 1.02i)T + (0.595 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.324 - 2.98i)T + (-12.6 - 2.79i)T^{2} \) |
| 17 | \( 1 + (6.60 + 1.83i)T + (14.5 + 8.76i)T^{2} \) |
| 19 | \( 1 + (-1.91 - 1.45i)T + (5.08 + 18.3i)T^{2} \) |
| 23 | \( 1 + (3.50 + 6.61i)T + (-12.9 + 19.0i)T^{2} \) |
| 29 | \( 1 + (-6.12 + 4.15i)T + (10.7 - 26.9i)T^{2} \) |
| 31 | \( 1 + (4.49 + 5.91i)T + (-8.29 + 29.8i)T^{2} \) |
| 37 | \( 1 + (0.431 + 1.96i)T + (-33.5 + 15.5i)T^{2} \) |
| 41 | \( 1 + (-2.48 - 1.31i)T + (23.0 + 33.9i)T^{2} \) |
| 43 | \( 1 + (-4.78 - 5.05i)T + (-2.32 + 42.9i)T^{2} \) |
| 47 | \( 1 + (7.93 - 3.67i)T + (30.4 - 35.8i)T^{2} \) |
| 53 | \( 1 + (-1.20 + 2.00i)T + (-24.8 - 46.8i)T^{2} \) |
| 61 | \( 1 + (-10.2 - 6.98i)T + (22.5 + 56.6i)T^{2} \) |
| 67 | \( 1 + (-1.31 + 5.95i)T + (-60.8 - 28.1i)T^{2} \) |
| 71 | \( 1 + (-0.638 - 1.37i)T + (-45.9 + 54.1i)T^{2} \) |
| 73 | \( 1 + (0.995 - 2.95i)T + (-58.1 - 44.1i)T^{2} \) |
| 79 | \( 1 + (-0.685 + 12.6i)T + (-78.5 - 8.54i)T^{2} \) |
| 83 | \( 1 + (1.16 - 7.10i)T + (-78.6 - 26.5i)T^{2} \) |
| 89 | \( 1 + (-5.81 - 8.57i)T + (-32.9 + 82.6i)T^{2} \) |
| 97 | \( 1 + (-0.296 - 0.879i)T + (-77.2 + 58.7i)T^{2} \) |
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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
−12.98027536803784495053911511872, −11.78115632843514938759821110811, −10.75403980814443386305411943141, −9.616015365590720591186273200821, −8.862077521967214937965089589008, −7.81478259950637058662041369897, −6.56423066196272909825552340221, −4.56489688065580652883377989492, −4.08478708336514296877518405020, −2.31629453776606873261672150839,
1.98820760568424444162415734359, 3.07541151270256120206636971915, 5.53288114190687951950491020605, 6.39174023961759914841650118310, 7.08903235111282604780629314321, 8.603955516814362771924227267618, 9.565394580967847338322343994238, 10.67158257803449912364604893641, 11.61486446833571485948175844938, 12.77604596434775102491550773592
