L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.707 − 0.707i)3-s + (0.809 − 0.587i)4-s + (−0.587 − 0.809i)5-s + (−0.891 − 0.453i)6-s + (−0.891 + 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (−0.809 − 0.587i)10-s + (−0.156 − 0.987i)11-s + (−0.987 − 0.156i)12-s + (0.453 − 0.891i)13-s + (−0.707 + 0.707i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + (0.987 − 0.156i)17-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (−0.707 − 0.707i)3-s + (0.809 − 0.587i)4-s + (−0.587 − 0.809i)5-s + (−0.891 − 0.453i)6-s + (−0.891 + 0.453i)7-s + (0.587 − 0.809i)8-s + i·9-s + (−0.809 − 0.587i)10-s + (−0.156 − 0.987i)11-s + (−0.987 − 0.156i)12-s + (0.453 − 0.891i)13-s + (−0.707 + 0.707i)14-s + (−0.156 + 0.987i)15-s + (0.309 − 0.951i)16-s + (0.987 − 0.156i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 41 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.661 - 0.749i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6729055799 - 1.491983522i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6729055799 - 1.491983522i\) |
\(L(1)\) |
\(\approx\) |
\(1.023766646 - 0.8004843659i\) |
\(L(1)\) |
\(\approx\) |
\(1.023766646 - 0.8004843659i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 41 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 3 | \( 1 + (-0.707 - 0.707i)T \) |
| 5 | \( 1 + (-0.587 - 0.809i)T \) |
| 7 | \( 1 + (-0.891 + 0.453i)T \) |
| 11 | \( 1 + (-0.156 - 0.987i)T \) |
| 13 | \( 1 + (0.453 - 0.891i)T \) |
| 17 | \( 1 + (0.987 - 0.156i)T \) |
| 19 | \( 1 + (0.453 + 0.891i)T \) |
| 23 | \( 1 + (-0.309 - 0.951i)T \) |
| 29 | \( 1 + (0.987 + 0.156i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (-0.809 + 0.587i)T \) |
| 43 | \( 1 + (0.951 - 0.309i)T \) |
| 47 | \( 1 + (-0.891 - 0.453i)T \) |
| 53 | \( 1 + (-0.987 - 0.156i)T \) |
| 59 | \( 1 + (0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.951 - 0.309i)T \) |
| 67 | \( 1 + (0.156 - 0.987i)T \) |
| 71 | \( 1 + (0.156 + 0.987i)T \) |
| 73 | \( 1 - iT \) |
| 79 | \( 1 + (0.707 + 0.707i)T \) |
| 83 | \( 1 + T \) |
| 89 | \( 1 + (-0.891 + 0.453i)T \) |
| 97 | \( 1 + (-0.156 + 0.987i)T \) |
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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
−34.68997971595070885116514726056, −33.71424207290313302459421022631, −32.89011131722217324851166934882, −31.71383052920746522826744682494, −30.49051818434461585650639293679, −29.35263563907020470536416551480, −28.09730181415714847078202563135, −26.41581777026054848137726805329, −25.80340858541707562041189675419, −23.64588223566431347508941500633, −23.04719070115530769550936848735, −22.16013592379341195103693129047, −20.93313596344063023802369098883, −19.475971553221859662940446500859, −17.540726594795404805124350945848, −16.134934394061784444567461934704, −15.43049324051335569871295678794, −14.04749245645968644191277009762, −12.33774713033189771653456639865, −11.25069234649182643277702543942, −9.90994124717835734784792967513, −7.29653243598609805153400975353, −6.23598321994723861335216780823, −4.45936508538561745617847093798, −3.29483036967315025829631670937,
0.874179625199304787850445905404, 3.240198581243420492737837083632, 5.23691293896699851396966147477, 6.26470247728448078138696245484, 8.08331957482792937141525805832, 10.42298347794919516037006233091, 11.97528660007914481180979820906, 12.58895093243244807943044395317, 13.783852158985861736252823392286, 15.82367041726060293470669545646, 16.514341942725312675384858610900, 18.655711274187246033663991072, 19.587920599331593496894609853535, 21.00385847079491592661595880903, 22.49840649978178221280150227820, 23.23826233711883285967640511069, 24.402826393907151675157399305323, 25.22959769037596414216454221872, 27.61533598573115453515537746090, 28.68081401588287377498735593786, 29.458303008898304810648104646119, 30.69991884172316337822247624571, 31.8988087549391011015803577952, 32.6853960812580782577032783529, 34.40112538164095658340871852742