L(s) = 1 | + (0.5 + 0.866i)3-s − i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)3-s − i·5-s + (−0.866 − 0.5i)7-s + (−0.5 + 0.866i)9-s + (−0.866 + 0.5i)11-s + (−0.866 + 0.5i)15-s + (0.5 − 0.866i)17-s + (−0.866 − 0.5i)19-s − i·21-s + (0.5 + 0.866i)23-s − 25-s − 27-s + (0.5 + 0.866i)29-s − i·31-s + (−0.866 − 0.5i)33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 104 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.999 + 0.0386i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01856619149 + 0.9615167475i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01856619149 + 0.9615167475i\) |
\(L(1)\) |
\(\approx\) |
\(0.7791197564 + 0.5038713153i\) |
\(L(1)\) |
\(\approx\) |
\(0.7791197564 + 0.5038713153i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 13 | \( 1 \) |
good | 3 | \( 1 + (0.5 + 0.866i)T \) |
| 5 | \( 1 - iT \) |
| 7 | \( 1 + (-0.866 - 0.5i)T \) |
| 11 | \( 1 + (-0.866 + 0.5i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 - 0.5i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (0.5 + 0.866i)T \) |
| 31 | \( 1 - iT \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.866 + 0.5i)T \) |
| 43 | \( 1 + (-0.5 + 0.866i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 - T \) |
| 59 | \( 1 + (0.866 + 0.5i)T \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 - 0.5i)T \) |
| 71 | \( 1 + (0.866 + 0.5i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 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
−28.893235135279136294963444254330, −28.38505334767881651528332732640, −26.78072388387718649242246469628, −25.61886188945199126078813679260, −24.96589469544259868752579550237, −23.89269737821990242769792511024, −23.12065185372176799694318017423, −21.46473324496791689374620116308, −20.578817675048993578450454594593, −19.34073722120516712554118467501, −18.79344315007894241742918867801, −17.37128885358502873616147758341, −16.27412345293212775829823189059, −15.11476659067948727405038563158, −13.65937306770214469249603370868, −12.75540443426274746047666034845, −12.16237166704201487090496576473, −10.25941512504828976271250982593, −8.78938358540149870700394433866, −8.19509989172774010019689309431, −6.57379040462667566767028529226, −5.43938193884212680833582736801, −3.53038191371194864398591723760, −2.07126954383236786744123005812, −0.36785347872879074769724840036,
2.613300710192517894419852067235, 3.52865722331341337910665181241, 5.02518686330200669968244721699, 6.69982896006202338419170819752, 7.846345911065288456471503359782, 9.503343595332689398761762661256, 10.24258445154991181840702515909, 11.23136234264449978724732589636, 13.05837320798410591555357894299, 14.07839180734847635945028534902, 15.19857230369041566048695901011, 15.95822015000871238828332547920, 17.22897876246934697269237145282, 18.65171966404113938413692231774, 19.58490874041101567317680340420, 20.65603330539025234159141701851, 21.71883427244563179715421538965, 22.691913956223188525108261458814, 23.470551600731477384864115498453, 25.47299359130401631047554471968, 25.86121118489044684664990423402, 26.797150869079932226345574636431, 27.72943885591666633698688290815, 29.0572198989982306158755374061, 30.01061140166958768438932955460