L(s) = 1 | + (0.736 − 0.676i)2-s + (0.809 + 0.587i)3-s + (0.0855 − 0.996i)4-s + (0.993 − 0.113i)6-s + (0.466 + 0.884i)7-s + (−0.610 − 0.791i)8-s + (0.309 + 0.951i)9-s + (0.654 − 0.755i)12-s + (0.921 + 0.389i)13-s + (0.941 + 0.336i)14-s + (−0.985 − 0.170i)16-s + (0.254 + 0.967i)17-s + (0.870 + 0.491i)18-s + (−0.362 − 0.931i)19-s + (−0.142 + 0.989i)21-s + ⋯ |
L(s) = 1 | + (0.736 − 0.676i)2-s + (0.809 + 0.587i)3-s + (0.0855 − 0.996i)4-s + (0.993 − 0.113i)6-s + (0.466 + 0.884i)7-s + (−0.610 − 0.791i)8-s + (0.309 + 0.951i)9-s + (0.654 − 0.755i)12-s + (0.921 + 0.389i)13-s + (0.941 + 0.336i)14-s + (−0.985 − 0.170i)16-s + (0.254 + 0.967i)17-s + (0.870 + 0.491i)18-s + (−0.362 − 0.931i)19-s + (−0.142 + 0.989i)21-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.995 - 0.0933i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.909888744 - 0.1360911034i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.909888744 - 0.1360911034i\) |
\(L(1)\) |
\(\approx\) |
\(2.051848346 - 0.2214027761i\) |
\(L(1)\) |
\(\approx\) |
\(2.051848346 - 0.2214027761i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.736 - 0.676i)T \) |
| 3 | \( 1 + (0.809 + 0.587i)T \) |
| 7 | \( 1 + (0.466 + 0.884i)T \) |
| 13 | \( 1 + (0.921 + 0.389i)T \) |
| 17 | \( 1 + (0.254 + 0.967i)T \) |
| 19 | \( 1 + (-0.362 - 0.931i)T \) |
| 23 | \( 1 + (0.142 + 0.989i)T \) |
| 29 | \( 1 + (-0.998 + 0.0570i)T \) |
| 31 | \( 1 + (0.516 - 0.856i)T \) |
| 37 | \( 1 + (-0.974 - 0.226i)T \) |
| 41 | \( 1 + (0.198 - 0.980i)T \) |
| 43 | \( 1 + (0.959 + 0.281i)T \) |
| 47 | \( 1 + (0.870 - 0.491i)T \) |
| 53 | \( 1 + (0.985 - 0.170i)T \) |
| 59 | \( 1 + (0.198 + 0.980i)T \) |
| 61 | \( 1 + (-0.736 - 0.676i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (0.774 - 0.633i)T \) |
| 73 | \( 1 + (-0.696 - 0.717i)T \) |
| 79 | \( 1 + (-0.564 - 0.825i)T \) |
| 83 | \( 1 + (-0.897 - 0.441i)T \) |
| 89 | \( 1 + (0.841 - 0.540i)T \) |
| 97 | \( 1 + (0.0285 + 0.999i)T \) |
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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
−23.1414018544423271676114833860, −22.67602173848828192384290390496, −21.1777483201842583433792164636, −20.70276260830671377452192308336, −20.13697127131730300896362999007, −18.804629750090249787401162906107, −18.08252984387026343035077396833, −17.18856264538281685808835088262, −16.2791323584798222513846566771, −15.39818547591899362671468571022, −14.37924797140599365815475335255, −14.02965603035934961868490066719, −13.16661293744989396814303071380, −12.44043731422448461556173358768, −11.41886493789870671837488737200, −10.28241249203326245487183924195, −8.896407506262158831362359525301, −8.17506969310144514183268314835, −7.40961509009940855962367349778, −6.665756799051073268145276786159, −5.619840439187193079448924566737, −4.35128095409044890544412807301, −3.59713940870686031991828986613, −2.59027712938657402843777019306, −1.208905471825322831327709868536,
1.62123860958457506032095158965, 2.37888112104043493473409921373, 3.494711104218104001061301816318, 4.22416891097632380198340672586, 5.28173735541511475715691003287, 6.08225477068659513974619761401, 7.53428452539942662657766219560, 8.828790155886365617889194684929, 9.19709947412843084487548927982, 10.45139836038498918300989349068, 11.10965048488026668371773842778, 12.0214524830089600710729358934, 13.11866789529318428275624694552, 13.727639561443832386583055837065, 14.70055090614528997998219602669, 15.312528161074000786904859245688, 15.87576560326393871108253324749, 17.26433260569890779486078747968, 18.52902317017002125134811478763, 19.12445041912733727620939637471, 19.87031043358381139611043660448, 20.94363332211437064609208167994, 21.24162257323259758937763250024, 21.96982902717905786643707999358, 22.81586495557052424577674588509