L(s) = 1 | + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)5-s + (−0.623 + 0.781i)8-s + (−0.988 + 0.149i)10-s + (0.222 − 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 − 0.997i)26-s + ⋯ |
L(s) = 1 | + (0.733 + 0.680i)2-s + (0.0747 + 0.997i)4-s + (−0.623 + 0.781i)5-s + (−0.623 + 0.781i)8-s + (−0.988 + 0.149i)10-s + (0.222 − 0.974i)11-s + (−0.733 − 0.680i)13-s + (−0.988 + 0.149i)16-s + (−0.0747 + 0.997i)17-s + (−0.5 − 0.866i)19-s + (−0.826 − 0.563i)20-s + (0.826 − 0.563i)22-s + (0.900 + 0.433i)23-s + (−0.222 − 0.974i)25-s + (−0.0747 − 0.997i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.700 - 0.713i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9000135190 - 0.3775012306i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9000135190 - 0.3775012306i\) |
\(L(1)\) |
\(\approx\) |
\(1.032194430 + 0.4599641075i\) |
\(L(1)\) |
\(\approx\) |
\(1.032194430 + 0.4599641075i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (0.733 + 0.680i)T \) |
| 5 | \( 1 + (-0.623 + 0.781i)T \) |
| 11 | \( 1 + (0.222 - 0.974i)T \) |
| 13 | \( 1 + (-0.733 - 0.680i)T \) |
| 17 | \( 1 + (-0.0747 + 0.997i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (0.900 + 0.433i)T \) |
| 29 | \( 1 + (-0.826 - 0.563i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.826 + 0.563i)T \) |
| 41 | \( 1 + (-0.365 - 0.930i)T \) |
| 43 | \( 1 + (0.365 - 0.930i)T \) |
| 47 | \( 1 + (0.733 + 0.680i)T \) |
| 53 | \( 1 + (-0.826 + 0.563i)T \) |
| 59 | \( 1 + (-0.365 + 0.930i)T \) |
| 61 | \( 1 + (0.0747 - 0.997i)T \) |
| 67 | \( 1 + (-0.5 - 0.866i)T \) |
| 71 | \( 1 + (0.900 + 0.433i)T \) |
| 73 | \( 1 + (0.955 + 0.294i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (0.733 - 0.680i)T \) |
| 89 | \( 1 + (0.733 - 0.680i)T \) |
| 97 | \( 1 + (-0.5 - 0.866i)T \) |
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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
−23.67756279318117881046013788974, −23.12333965681254327089418946094, −22.30232311259381355632665609143, −21.26320264908947005280146967537, −20.516875990278354478608051158, −19.887633200743054089581227473652, −19.08774862660008346107143682754, −18.14560465208870561851519966224, −16.82834116333640458863782964910, −16.085929228562753915052616930150, −14.92677211205618144113767690909, −14.43853429452843850358082089487, −13.12622422704885139594588582974, −12.45895234229568303994863215715, −11.808772257167724742938179087709, −10.88231639773702055550159413225, −9.64666678241797351203284894958, −9.0479070101329722115259326028, −7.56253228468914264341615477343, −6.632886690638176395700697258030, −5.16389568457090340369296039307, −4.6082858832165975923139158246, −3.64996943826098123778501351851, −2.31199144520926720260348622399, −1.21795848730741028534465459951,
0.20465308205764386620006210571, 2.46971247788699383550507350600, 3.398177560419605797907462945834, 4.265685330712423764443974002858, 5.5316409596094097442798612672, 6.397508320409849219774360237768, 7.37621778344702172961477294283, 8.09266114874203853921369081553, 9.17220165295960931173026514407, 10.75481565842386577413175991592, 11.35471205120333848310496411316, 12.4441196960372956049017268954, 13.299517368401162172107309180414, 14.247872319066419084021228271304, 15.22379157525077711661835856507, 15.43422827719517925296049626944, 16.85399148145870921966779745471, 17.32135023512599986960043873377, 18.612824544874491240429208086407, 19.39014848632873886794016701768, 20.3882326600407389166730632561, 21.6329198812513957532038179219, 22.08645716000516973131559302580, 22.87811060099727498323573222710, 23.87206680023095840320376372998