L(s) = 1 | + (0.0275 − 0.999i)2-s + (0.904 − 0.426i)3-s + (−0.998 − 0.0550i)4-s + (−0.754 + 0.656i)5-s + (−0.401 − 0.915i)6-s + (0.245 − 0.969i)7-s + (−0.0825 + 0.996i)8-s + (0.635 − 0.771i)9-s + (0.635 + 0.771i)10-s + (0.350 + 0.936i)11-s + (−0.926 + 0.376i)12-s + (−0.298 + 0.954i)13-s + (−0.962 − 0.272i)14-s + (−0.401 + 0.915i)15-s + (0.993 + 0.110i)16-s + (0.993 + 0.110i)17-s + ⋯ |
L(s) = 1 | + (0.0275 − 0.999i)2-s + (0.904 − 0.426i)3-s + (−0.998 − 0.0550i)4-s + (−0.754 + 0.656i)5-s + (−0.401 − 0.915i)6-s + (0.245 − 0.969i)7-s + (−0.0825 + 0.996i)8-s + (0.635 − 0.771i)9-s + (0.635 + 0.771i)10-s + (0.350 + 0.936i)11-s + (−0.926 + 0.376i)12-s + (−0.298 + 0.954i)13-s + (−0.962 − 0.272i)14-s + (−0.401 + 0.915i)15-s + (0.993 + 0.110i)16-s + (0.993 + 0.110i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 571 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.166 - 0.986i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.053245355 - 1.245497103i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.053245355 - 1.245497103i\) |
\(L(1)\) |
\(\approx\) |
\(1.044603081 - 0.7123792205i\) |
\(L(1)\) |
\(\approx\) |
\(1.044603081 - 0.7123792205i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 571 | \( 1 \) |
good | 2 | \( 1 + (0.0275 - 0.999i)T \) |
| 3 | \( 1 + (0.904 - 0.426i)T \) |
| 5 | \( 1 + (-0.754 + 0.656i)T \) |
| 7 | \( 1 + (0.245 - 0.969i)T \) |
| 11 | \( 1 + (0.350 + 0.936i)T \) |
| 13 | \( 1 + (-0.298 + 0.954i)T \) |
| 17 | \( 1 + (0.993 + 0.110i)T \) |
| 19 | \( 1 + (0.904 - 0.426i)T \) |
| 23 | \( 1 + (-0.0825 - 0.996i)T \) |
| 29 | \( 1 + (0.975 - 0.218i)T \) |
| 31 | \( 1 + (-0.401 - 0.915i)T \) |
| 37 | \( 1 + (-0.298 - 0.954i)T \) |
| 41 | \( 1 + (-0.754 + 0.656i)T \) |
| 43 | \( 1 + (0.635 + 0.771i)T \) |
| 47 | \( 1 + (-0.592 + 0.805i)T \) |
| 53 | \( 1 + (0.0275 + 0.999i)T \) |
| 59 | \( 1 + (0.245 - 0.969i)T \) |
| 61 | \( 1 + (0.851 + 0.523i)T \) |
| 67 | \( 1 + (0.851 - 0.523i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + (0.975 + 0.218i)T \) |
| 79 | \( 1 + (0.451 - 0.892i)T \) |
| 83 | \( 1 + (-0.592 + 0.805i)T \) |
| 89 | \( 1 + (0.993 + 0.110i)T \) |
| 97 | \( 1 + (-0.998 - 0.0550i)T \) |
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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
−23.78342244110583994031413172572, −22.67009850038261795452004239450, −21.840703835693418566186368415294, −21.12979893844183689023328040723, −20.07451217671817988300300867540, −19.20096324605342284678136687741, −18.60274883088423615374857755817, −17.47000328066146178755228662112, −16.34222553695652322705921171414, −15.875216254977469146226290572936, −15.17606700851138966389591677320, −14.38433213790385574828867742071, −13.588762174579468392681427041389, −12.544367342362966535473946127120, −11.758825264055508585704249684133, −10.19500985781158020819371501287, −9.30563860516579027334782800736, −8.39357413421810110788334290129, −8.139212305613825284289635375208, −7.09433184901334520336442125396, −5.390877186960085864637335533112, −5.21388681242004661069098129505, −3.704079962848468735408593220357, −3.16226684048440196213003214820, −1.20218099243068383784634645676,
0.96382212902182069724312888122, 2.10219219486848247436620996749, 3.10838138837256723648573521834, 4.012674819737421244611740739120, 4.62393398889857816850058383733, 6.62505735259283907654149735938, 7.48813049632702754360822275930, 8.14028595597195556436571694114, 9.418647034145661236258266949437, 10.01613852751048617886507653593, 11.07684026904181494655766132442, 11.97937463298283934512872277420, 12.62461488136488045494937147650, 13.827837500805416179984118132, 14.34228812873850667224181843009, 14.93429412536967141005703072750, 16.33257696731268715273908969333, 17.53257153313290532203348750406, 18.31000926877830983888236321715, 19.12137733985504221647232932562, 19.715283395861649534033546800933, 20.3647951413191887040878747914, 21.05404855789703518619359361534, 22.14964531253499387355251042475, 23.064537607832138884292340435517