L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.973 − 0.230i)3-s + (−0.939 + 0.342i)4-s + (0.835 − 0.549i)5-s + (−0.396 − 0.918i)6-s + (−0.0581 − 0.998i)7-s + (0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.835 + 0.549i)12-s + (0.0581 + 0.998i)13-s + (−0.973 + 0.230i)14-s + (0.686 − 0.727i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯ |
L(s) = 1 | + (−0.173 − 0.984i)2-s + (0.973 − 0.230i)3-s + (−0.939 + 0.342i)4-s + (0.835 − 0.549i)5-s + (−0.396 − 0.918i)6-s + (−0.0581 − 0.998i)7-s + (0.5 + 0.866i)8-s + (0.893 − 0.448i)9-s + (−0.686 − 0.727i)10-s + (−0.686 + 0.727i)11-s + (−0.835 + 0.549i)12-s + (0.0581 + 0.998i)13-s + (−0.973 + 0.230i)14-s + (0.686 − 0.727i)15-s + (0.766 − 0.642i)16-s + (−0.766 + 0.642i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.993 - 0.112i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1161420114 - 2.049698214i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1161420114 - 2.049698214i\) |
\(L(1)\) |
\(\approx\) |
\(0.9450015311 - 0.9332438575i\) |
\(L(1)\) |
\(\approx\) |
\(0.9450015311 - 0.9332438575i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 + (-0.173 - 0.984i)T \) |
| 3 | \( 1 + (0.973 - 0.230i)T \) |
| 5 | \( 1 + (0.835 - 0.549i)T \) |
| 7 | \( 1 + (-0.0581 - 0.998i)T \) |
| 11 | \( 1 + (-0.686 + 0.727i)T \) |
| 13 | \( 1 + (0.0581 + 0.998i)T \) |
| 17 | \( 1 + (-0.766 + 0.642i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.766 - 0.642i)T \) |
| 29 | \( 1 + (0.286 - 0.957i)T \) |
| 31 | \( 1 + (0.835 + 0.549i)T \) |
| 41 | \( 1 + (-0.5 - 0.866i)T \) |
| 43 | \( 1 + (0.939 + 0.342i)T \) |
| 47 | \( 1 + (0.597 + 0.802i)T \) |
| 53 | \( 1 + (-0.0581 - 0.998i)T \) |
| 59 | \( 1 + (0.686 - 0.727i)T \) |
| 61 | \( 1 + (-0.396 - 0.918i)T \) |
| 67 | \( 1 + (0.893 - 0.448i)T \) |
| 71 | \( 1 + (0.173 - 0.984i)T \) |
| 73 | \( 1 + (-0.686 + 0.727i)T \) |
| 79 | \( 1 + (0.835 + 0.549i)T \) |
| 83 | \( 1 + (-0.686 - 0.727i)T \) |
| 89 | \( 1 + (-0.597 + 0.802i)T \) |
| 97 | \( 1 + (0.0581 + 0.998i)T \) |
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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
−18.47986449375469278603815822119, −18.29356018191318245261444153721, −17.57487003848567559990132530290, −16.588240406321029800706589653293, −15.790068900813387699495337870852, −15.488734300586406552468516189592, −14.7606818219226214447428729840, −14.18231017350555622595518980515, −13.4612055885425301227819806971, −13.13825571236803644314228729800, −12.17215299992229657038586649160, −10.879201295525372888237108058212, −10.19364216381047768219522029892, −9.71297931255531698906359331796, −8.88364836045469295084208727111, −8.41713968971122516485776608216, −7.73090614192854360767252203036, −6.97018232938757482951180212368, −5.96833513939735776155329922174, −5.63252671803976121271894146250, −4.819926817780123541192835347889, −3.72822722516237412735983644886, −2.90606138897853674942703780129, −2.328351161796911435791350586431, −1.22756947507189651951379973838,
0.51873374849558356748668260232, 1.52518922631434824840866641323, 2.16724297230155291434976081937, 2.64175528726232607809099687470, 3.8616311198010465934699903242, 4.40778401151337660185286894893, 4.90249573918101439001130248255, 6.339962262791927953058113767935, 6.99521953918169323688311171037, 7.98369035805961391532056711804, 8.50702559311617400402861308458, 9.284145750135702546310723397998, 9.81399771262042828409361153034, 10.37746599635795815494914527221, 11.09766951834479717255651179626, 12.212552558075875361315601276136, 12.78716886502558980350178241010, 13.303936969143453403512191571, 13.96394073316888210654893633609, 14.20659169280336468502742893484, 15.376971472841032281447060231143, 16.14506198581169358286489505076, 17.1537742999232673326785232596, 17.585829844519632496062964380984, 18.130212453501847394793670315553