L(s) = 1 | + (0.989 + 0.142i)2-s + (0.959 + 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.841 + 0.540i)7-s + (0.909 + 0.415i)8-s + (−0.540 − 0.841i)10-s + (−0.989 + 0.142i)11-s + (−0.841 + 0.540i)13-s + (0.755 + 0.654i)14-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + (0.281 − 0.959i)19-s + (−0.415 − 0.909i)20-s − 22-s + (−0.142 + 0.989i)25-s + (−0.909 + 0.415i)26-s + ⋯ |
L(s) = 1 | + (0.989 + 0.142i)2-s + (0.959 + 0.281i)4-s + (−0.654 − 0.755i)5-s + (0.841 + 0.540i)7-s + (0.909 + 0.415i)8-s + (−0.540 − 0.841i)10-s + (−0.989 + 0.142i)11-s + (−0.841 + 0.540i)13-s + (0.755 + 0.654i)14-s + (0.841 + 0.540i)16-s + (0.281 + 0.959i)17-s + (0.281 − 0.959i)19-s + (−0.415 − 0.909i)20-s − 22-s + (−0.142 + 0.989i)25-s + (−0.909 + 0.415i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.489 + 0.871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.404255780 + 1.406801838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.404255780 + 1.406801838i\) |
\(L(1)\) |
\(\approx\) |
\(1.773809338 + 0.3452450869i\) |
\(L(1)\) |
\(\approx\) |
\(1.773809338 + 0.3452450869i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 23 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (0.989 + 0.142i)T \) |
| 5 | \( 1 + (-0.654 - 0.755i)T \) |
| 7 | \( 1 + (0.841 + 0.540i)T \) |
| 11 | \( 1 + (-0.989 + 0.142i)T \) |
| 13 | \( 1 + (-0.841 + 0.540i)T \) |
| 17 | \( 1 + (0.281 + 0.959i)T \) |
| 19 | \( 1 + (0.281 - 0.959i)T \) |
| 31 | \( 1 + (0.909 + 0.415i)T \) |
| 37 | \( 1 + (0.755 + 0.654i)T \) |
| 41 | \( 1 + (0.755 - 0.654i)T \) |
| 43 | \( 1 + (-0.909 + 0.415i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + (-0.841 - 0.540i)T \) |
| 59 | \( 1 + (-0.841 + 0.540i)T \) |
| 61 | \( 1 + (0.909 + 0.415i)T \) |
| 67 | \( 1 + (0.142 - 0.989i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.281 + 0.959i)T \) |
| 79 | \( 1 + (0.540 + 0.841i)T \) |
| 83 | \( 1 + (0.654 - 0.755i)T \) |
| 89 | \( 1 + (0.909 - 0.415i)T \) |
| 97 | \( 1 + (-0.755 + 0.654i)T \) |
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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
−20.10120260074440473464339389759, −19.19611995780285519683774830193, −18.480868890726124371707428923540, −17.75131316784773353570526440852, −16.665087144358090420753502593553, −16.02246003191322997298437229016, −15.19605001249713535225725613736, −14.68515888394045924154702539701, −14.0270040471972930409157380188, −13.35588871711756126258158107937, −12.356280804725324886709811492261, −11.760134020003242997601884626869, −11.059488260351209157809965216349, −10.405117298800673873413838435476, −9.79442843213864257509202739926, −7.994801672042518964066632830090, −7.75672593809113442909544585252, −7.04661870642928111868492272713, −6.013834844353127423186675924454, −5.10906224669851186347396330554, −4.54614693790946218899369037309, −3.557608884530547644659580788301, −2.846192258532533535851299438279, −2.06858968852810046254397657466, −0.68314993955848650597268591992,
1.247039142137229780725140831488, 2.244744259398548618391777537010, 3.0192190603788446191611143160, 4.20781933844359254112978556952, 4.81620299835358738408436546197, 5.259912031307925038694360834884, 6.261083178509886072689672979541, 7.33400186877740364892430544145, 7.94153216549961617090438446479, 8.544112801608788066396593462154, 9.65791150937062779344451011677, 10.75123368233250232074847191118, 11.45124363611977561767192870438, 12.08022891866514737650048103068, 12.69465462774404204341872689358, 13.35588662197783802656185024888, 14.32067800664380153325306870025, 14.99245464723251518876983726689, 15.57462229120968030508742913941, 16.16947010354494165180071400085, 17.10014059031990594688763000635, 17.629395983814962207147228734562, 18.82788703983958789988629106729, 19.52675796930860235048783654568, 20.231880148948888836706266732424