L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 − 0.781i)8-s + (0.623 − 0.781i)10-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)26-s + ⋯ |
L(s) = 1 | + (0.222 + 0.974i)2-s + (−0.900 + 0.433i)4-s + (−0.623 − 0.781i)5-s + (−0.623 − 0.781i)8-s + (0.623 − 0.781i)10-s + (0.222 + 0.974i)11-s + (−0.222 − 0.974i)13-s + (0.623 − 0.781i)16-s + (0.900 + 0.433i)17-s + 19-s + (0.900 + 0.433i)20-s + (−0.900 + 0.433i)22-s + (0.900 − 0.433i)23-s + (−0.222 + 0.974i)25-s + (0.900 − 0.433i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.718 + 0.695i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.478938088 + 0.5987556728i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.478938088 + 0.5987556728i\) |
\(L(1)\) |
\(\approx\) |
\(1.003327853 + 0.3800528324i\) |
\(L(1)\) |
\(\approx\) |
\(1.003327853 + 0.3800528324i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.222 + 0.974i)T \) |
| 5 | \( 1 + (-0.623 - 0.781i)T \) |
| 11 | \( 1 + (0.222 + 0.974i)T \) |
| 13 | \( 1 + (-0.222 - 0.974i)T \) |
| 17 | \( 1 + (0.900 + 0.433i)T \) |
| 19 | \( 1 + T \) |
| 23 | \( 1 + (0.900 - 0.433i)T \) |
| 29 | \( 1 + (0.900 + 0.433i)T \) |
| 31 | \( 1 + T \) |
| 37 | \( 1 + (-0.900 - 0.433i)T \) |
| 41 | \( 1 + (-0.623 - 0.781i)T \) |
| 43 | \( 1 + (0.623 - 0.781i)T \) |
| 47 | \( 1 + (0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.900 - 0.433i)T \) |
| 59 | \( 1 + (-0.623 + 0.781i)T \) |
| 61 | \( 1 + (-0.900 - 0.433i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (0.900 - 0.433i)T \) |
| 73 | \( 1 + (-0.222 + 0.974i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.222 - 0.974i)T \) |
| 89 | \( 1 + (0.222 - 0.974i)T \) |
| 97 | \( 1 + T \) |
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\(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−27.813439594355242470853958797773, −26.91016550693761459884901266342, −26.35907149037805326354780241431, −24.66919278320577757525482532655, −23.52730628540947174129602741521, −22.80393343439968959539113903266, −21.78387401944925430754985488727, −21.01948915907002659690192631541, −19.66083230707708641794017141475, −19.02146278113624128634391195166, −18.26235564353760807347158410726, −16.83327122407688631027664725346, −15.4990184177812365696438115848, −14.24999061308258221368653099743, −13.67918295326423554920781688149, −11.978661958448186145797792097650, −11.537868056096666380321733298958, −10.3922802921330239958794522488, −9.28183694570767220136253817630, −7.979250921854328853454207852521, −6.51995898179697726570854083606, −5.02801318467265742287022522755, −3.64983625301675173593742337904, −2.790106746523592839056708943223, −0.964643417684805153191457609405,
0.83620572290159702921690193524, 3.34515164728306834772091759657, 4.64629362837759255068539074892, 5.49707169654080879575190002091, 7.07882453602700507997186696347, 7.94139954370942916281976527514, 8.98477978625934364592443469354, 10.18790418252490913039514447957, 12.14429637867857594693190327223, 12.631428457555186039753910938046, 13.95990174646749091961925303292, 15.13334229282943936957088726740, 15.80389203092996956253944793781, 16.944859382781929936782642876284, 17.667743541604200344085932381882, 18.98001519820438345308284594974, 20.16567899313917280908791945518, 21.15677853448402173995170434860, 22.625903316306262042638887269336, 23.111435257746167759249202763510, 24.266236786729656767721372557998, 24.9764257004706649695958565614, 25.88175497578801182520526029095, 27.1742596387447925920985819592, 27.70577462095759108548826507712