L(s) = 1 | + (−0.149 − 0.988i)2-s + (−0.955 + 0.294i)4-s + (0.433 + 0.900i)8-s + (0.988 − 0.149i)11-s + (0.930 + 0.365i)13-s + (0.826 − 0.563i)16-s + (0.974 − 0.222i)17-s − 19-s + (−0.294 − 0.955i)22-s + (−0.294 − 0.955i)23-s + (0.222 − 0.974i)26-s + (0.955 + 0.294i)29-s + (−0.5 − 0.866i)31-s + (−0.680 − 0.733i)32-s + (−0.365 − 0.930i)34-s + ⋯ |
L(s) = 1 | + (−0.149 − 0.988i)2-s + (−0.955 + 0.294i)4-s + (0.433 + 0.900i)8-s + (0.988 − 0.149i)11-s + (0.930 + 0.365i)13-s + (0.826 − 0.563i)16-s + (0.974 − 0.222i)17-s − 19-s + (−0.294 − 0.955i)22-s + (−0.294 − 0.955i)23-s + (0.222 − 0.974i)26-s + (0.955 + 0.294i)29-s + (−0.5 − 0.866i)31-s + (−0.680 − 0.733i)32-s + (−0.365 − 0.930i)34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0590 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2205 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0590 - 0.998i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.100411088 - 1.037254807i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.100411088 - 1.037254807i\) |
\(L(1)\) |
\(\approx\) |
\(0.9081828245 - 0.4783707786i\) |
\(L(1)\) |
\(\approx\) |
\(0.9081828245 - 0.4783707786i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.149 - 0.988i)T \) |
| 11 | \( 1 + (0.988 - 0.149i)T \) |
| 13 | \( 1 + (0.930 + 0.365i)T \) |
| 17 | \( 1 + (0.974 - 0.222i)T \) |
| 19 | \( 1 - T \) |
| 23 | \( 1 + (-0.294 - 0.955i)T \) |
| 29 | \( 1 + (0.955 + 0.294i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (-0.974 + 0.222i)T \) |
| 41 | \( 1 + (-0.826 - 0.563i)T \) |
| 43 | \( 1 + (0.563 + 0.826i)T \) |
| 47 | \( 1 + (-0.149 - 0.988i)T \) |
| 53 | \( 1 + (0.974 + 0.222i)T \) |
| 59 | \( 1 + (0.0747 + 0.997i)T \) |
| 61 | \( 1 + (0.955 + 0.294i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.222 - 0.974i)T \) |
| 73 | \( 1 + (0.781 + 0.623i)T \) |
| 79 | \( 1 + (0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.930 - 0.365i)T \) |
| 89 | \( 1 + (0.623 - 0.781i)T \) |
| 97 | \( 1 + (0.866 + 0.5i)T \) |
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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
−19.52843147002967723522405767002, −19.2276324385819660114495930752, −18.27852592365038497379742208688, −17.5901471836789354826046077132, −17.05235169483341160821361994122, −16.2819199630243078413158140878, −15.61220287608603595595191944181, −14.925538051645754680218023206977, −14.17340751790618143463912532201, −13.6574500305380933409577697255, −12.69940510183358875826140352043, −12.065301971086424238344509117131, −10.945868117844893494545410670122, −10.188202443918788400832695911530, −9.42568052409018548362718225688, −8.598186189436225909500572226112, −8.11029780232661208705388633309, −7.1105080113397306020302994102, −6.44815047720310435564432146353, −5.7545094585540797465560566819, −4.95333308895012283880728828420, −3.9054338339352453208629927842, −3.42586032078780425578882861727, −1.77438188369942216900761982118, −0.9218259864565986067187545652,
0.72841312530450820277471568341, 1.60343955797374106739285583080, 2.47963641019405148214253031716, 3.56831486670250169462615945715, 4.02954468811575669693493162136, 4.96770100185142641218381870247, 5.989283503612216544252183487523, 6.7756730248058513588097900527, 7.92008354618849420243658951079, 8.72424969787357853831056302709, 9.119228794971316942606559137283, 10.25473078915664583435125953875, 10.60420712329444314306236886836, 11.71197794479165094659909207766, 11.985789265545009933045543472310, 12.92012597930433624348823529081, 13.65852368376718113001348102771, 14.33571313626615726185379684881, 14.963374294880464821867677840948, 16.257513032200684052323062911745, 16.75167539385672313867767022301, 17.51350256232992180333745039493, 18.3647479401307608547015101457, 18.904297726539259471671206607968, 19.52602977654603869370644988382