L(s) = 1 | + (−0.990 + 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (−0.334 + 0.942i)5-s + (−0.917 − 0.398i)6-s + (−0.0682 − 0.997i)7-s + (−0.917 + 0.398i)8-s + (0.460 + 0.887i)9-s + (0.203 − 0.979i)10-s + (0.682 + 0.730i)11-s + (0.962 + 0.269i)12-s + (−0.576 + 0.816i)13-s + (0.203 + 0.979i)14-s + (−0.775 + 0.631i)15-s + (0.854 − 0.519i)16-s + (0.682 − 0.730i)17-s + ⋯ |
L(s) = 1 | + (−0.990 + 0.136i)2-s + (0.854 + 0.519i)3-s + (0.962 − 0.269i)4-s + (−0.334 + 0.942i)5-s + (−0.917 − 0.398i)6-s + (−0.0682 − 0.997i)7-s + (−0.917 + 0.398i)8-s + (0.460 + 0.887i)9-s + (0.203 − 0.979i)10-s + (0.682 + 0.730i)11-s + (0.962 + 0.269i)12-s + (−0.576 + 0.816i)13-s + (0.203 + 0.979i)14-s + (−0.775 + 0.631i)15-s + (0.854 − 0.519i)16-s + (0.682 − 0.730i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.600 + 0.799i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6234525662 + 0.3115258406i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6234525662 + 0.3115258406i\) |
\(L(1)\) |
\(\approx\) |
\(0.7824977724 + 0.2481416599i\) |
\(L(1)\) |
\(\approx\) |
\(0.7824977724 + 0.2481416599i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 47 | \( 1 \) |
good | 2 | \( 1 + (-0.990 + 0.136i)T \) |
| 3 | \( 1 + (0.854 + 0.519i)T \) |
| 5 | \( 1 + (-0.334 + 0.942i)T \) |
| 7 | \( 1 + (-0.0682 - 0.997i)T \) |
| 11 | \( 1 + (0.682 + 0.730i)T \) |
| 13 | \( 1 + (-0.576 + 0.816i)T \) |
| 17 | \( 1 + (0.682 - 0.730i)T \) |
| 19 | \( 1 + (-0.334 - 0.942i)T \) |
| 23 | \( 1 + (-0.990 - 0.136i)T \) |
| 29 | \( 1 + (-0.576 - 0.816i)T \) |
| 31 | \( 1 + (0.854 - 0.519i)T \) |
| 37 | \( 1 + (0.203 - 0.979i)T \) |
| 41 | \( 1 + (-0.917 - 0.398i)T \) |
| 43 | \( 1 + (0.962 - 0.269i)T \) |
| 53 | \( 1 + (-0.917 - 0.398i)T \) |
| 59 | \( 1 + (0.962 + 0.269i)T \) |
| 61 | \( 1 + (0.203 + 0.979i)T \) |
| 67 | \( 1 + (-0.0682 + 0.997i)T \) |
| 71 | \( 1 + (-0.990 - 0.136i)T \) |
| 73 | \( 1 + (0.460 - 0.887i)T \) |
| 79 | \( 1 + (-0.775 + 0.631i)T \) |
| 83 | \( 1 + (0.682 + 0.730i)T \) |
| 89 | \( 1 + (-0.334 + 0.942i)T \) |
| 97 | \( 1 + (0.854 + 0.519i)T \) |
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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
−34.4933725910660864334356000636, −32.47437850939068819905582785931, −31.656563406998100355936940068524, −30.267528754406058897015102250681, −29.25262144357966816126281198640, −27.934909062948030008439832279568, −27.11939953837282503256882364133, −25.57297451259322960366899909172, −24.85195254758152896999547033893, −24.00868573799488640640385228586, −21.65507325601032396920666083368, −20.47178335508733962558901714742, −19.490857670016343568344119465320, −18.67596574755327393629109258425, −17.23461435620308207547409580653, −15.90839622784319245313040185462, −14.69576638469379737343893532285, −12.66809464052774675917273891607, −11.94172009573656985723695142601, −9.80654029397465976921071332171, −8.59628621865952055529311624089, −7.96092007561691093432527316199, −6.03914891580440251031353973123, −3.32623649267657408511531242475, −1.56036451724587251375148228121,
2.36371859092513967007418106284, 4.06222512716185316106117853085, 6.886210709753775509260096032261, 7.70207314118078666955637143326, 9.43359320885263024097602787100, 10.27432970829019054405040063030, 11.62055785865565846061478326369, 14.04122654019958219092778512493, 14.93821880729305429043895152879, 16.19153415481042200215457886760, 17.4625080644349846839914976261, 19.03015535030772702149182416785, 19.747952021490490115827127839102, 20.84003663135223380708201515183, 22.39749095625134947776423602565, 23.985001512722880958355654466545, 25.44565847628876872593625211400, 26.326301261868936959284596182, 26.96561387828985011596060971077, 28.060026684939245341517417592927, 29.85226970938329816879232134817, 30.46185315074404805324402044431, 32.11801176828295068127557218718, 33.43371607823650382814174122716, 34.061964879296941700148271210394