L(s) = 1 | + (0.909 + 0.415i)3-s + (0.540 + 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (0.281 + 0.959i)27-s + (0.959 + 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.540 + 0.841i)33-s + (−0.755 + 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯ |
L(s) = 1 | + (0.909 + 0.415i)3-s + (0.540 + 0.841i)7-s + (0.654 + 0.755i)9-s + (−0.142 + 0.989i)11-s + (0.540 − 0.841i)13-s + (−0.281 + 0.959i)17-s + (0.959 − 0.281i)19-s + (0.142 + 0.989i)21-s + (0.281 + 0.959i)27-s + (0.959 + 0.281i)29-s + (−0.415 − 0.909i)31-s + (−0.540 + 0.841i)33-s + (−0.755 + 0.654i)37-s + (0.841 − 0.540i)39-s + (−0.654 + 0.755i)41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.100 + 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.075160578 + 2.294194800i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.075160578 + 2.294194800i\) |
\(L(1)\) |
\(\approx\) |
\(1.502994010 + 0.6046033493i\) |
\(L(1)\) |
\(\approx\) |
\(1.502994010 + 0.6046033493i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 23 | \( 1 \) |
good | 3 | \( 1 + (0.909 + 0.415i)T \) |
| 7 | \( 1 + (0.540 + 0.841i)T \) |
| 11 | \( 1 + (-0.142 + 0.989i)T \) |
| 13 | \( 1 + (0.540 - 0.841i)T \) |
| 17 | \( 1 + (-0.281 + 0.959i)T \) |
| 19 | \( 1 + (0.959 - 0.281i)T \) |
| 29 | \( 1 + (0.959 + 0.281i)T \) |
| 31 | \( 1 + (-0.415 - 0.909i)T \) |
| 37 | \( 1 + (-0.755 + 0.654i)T \) |
| 41 | \( 1 + (-0.654 + 0.755i)T \) |
| 43 | \( 1 + (-0.909 - 0.415i)T \) |
| 47 | \( 1 - iT \) |
| 53 | \( 1 + (0.540 + 0.841i)T \) |
| 59 | \( 1 + (0.841 + 0.540i)T \) |
| 61 | \( 1 + (-0.415 - 0.909i)T \) |
| 67 | \( 1 + (0.989 - 0.142i)T \) |
| 71 | \( 1 + (0.142 + 0.989i)T \) |
| 73 | \( 1 + (0.281 + 0.959i)T \) |
| 79 | \( 1 + (-0.841 - 0.540i)T \) |
| 83 | \( 1 + (-0.755 + 0.654i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 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
−23.7548265619494951420085130472, −22.78349250355280771581092206875, −21.45408082479475533916766119500, −20.882487271777941548426976225735, −20.07685639631472010050064554668, −19.2649495583602416100998124958, −18.39729470642774212303326421003, −17.69889502613505596556340981581, −16.38964450348185331432887919025, −15.79574067757808030932471596175, −14.40464505530862979972826268561, −13.90873216722327232065374201737, −13.377852313524190169069096131632, −12.03621965853083619108696986155, −11.19549270954506153288814739089, −10.10731373551693217225637134716, −9.02014210572013181440979622873, −8.28845519092124123060650064133, −7.322864814775870508904674077001, −6.563460034308665041971340926150, −5.103580738152955820529819547691, −3.907588910089943924166178752420, −3.07739831803254281994418417692, −1.7269155380701338633618325107, −0.731543250782931852127000993934,
1.49780406234697498529468444783, 2.48537136429664545361568022594, 3.52631947076914576955872835558, 4.69036732752203201513075889824, 5.540026475177741546254848686484, 6.97830798389354392945664133240, 8.10342202869144714325909295681, 8.622146826449517813950401815922, 9.7285889892564779309575932209, 10.47070620707091756383263966650, 11.639941696110181375818093584019, 12.69356858556870083658000524636, 13.50284914309123535385082300475, 14.582437769755373631424972353492, 15.31709550490643525234007389483, 15.71340124762643288081087811401, 17.10610786810857303822654381061, 18.1071007886967491446032285538, 18.72581715123565146909507639947, 20.03636015790103776699395836198, 20.317623148206796965695681101423, 21.41142350110470694047923529725, 21.99193518356820512884751789014, 23.01175955630844998050298919576, 24.15407952771872110466260793407