L(s) = 1 | − 2-s + (−0.835 − 0.549i)3-s + 4-s + (−0.597 − 0.802i)5-s + (0.835 + 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.835 − 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.0581 + 0.998i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
L(s) = 1 | − 2-s + (−0.835 − 0.549i)3-s + 4-s + (−0.597 − 0.802i)5-s + (0.835 + 0.549i)6-s + (0.396 − 0.918i)7-s − 8-s + (0.396 + 0.918i)9-s + (0.597 + 0.802i)10-s + (0.597 − 0.802i)11-s + (−0.835 − 0.549i)12-s + (−0.597 − 0.802i)13-s + (−0.396 + 0.918i)14-s + (0.0581 + 0.998i)15-s + 16-s + (0.5 + 0.866i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.928 - 0.371i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5328419087 - 0.1027527945i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5328419087 - 0.1027527945i\) |
\(L(1)\) |
\(\approx\) |
\(0.4645385128 - 0.1924062819i\) |
\(L(1)\) |
\(\approx\) |
\(0.4645385128 - 0.1924062819i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 37 | \( 1 \) |
| 109 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.835 - 0.549i)T \) |
| 5 | \( 1 + (-0.597 - 0.802i)T \) |
| 7 | \( 1 + (0.396 - 0.918i)T \) |
| 11 | \( 1 + (0.597 - 0.802i)T \) |
| 13 | \( 1 + (-0.597 - 0.802i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (-0.173 - 0.984i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.396 + 0.918i)T \) |
| 31 | \( 1 + (0.286 - 0.957i)T \) |
| 41 | \( 1 + (0.766 + 0.642i)T \) |
| 43 | \( 1 + (-0.173 + 0.984i)T \) |
| 47 | \( 1 + (-0.835 + 0.549i)T \) |
| 53 | \( 1 + (0.893 - 0.448i)T \) |
| 59 | \( 1 + (0.0581 + 0.998i)T \) |
| 61 | \( 1 + (0.686 - 0.727i)T \) |
| 67 | \( 1 + (-0.835 + 0.549i)T \) |
| 71 | \( 1 + T \) |
| 73 | \( 1 + (0.396 + 0.918i)T \) |
| 79 | \( 1 + (0.0581 + 0.998i)T \) |
| 83 | \( 1 + (0.973 + 0.230i)T \) |
| 89 | \( 1 + (0.993 - 0.116i)T \) |
| 97 | \( 1 + (0.286 + 0.957i)T \) |
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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
−18.48138442684402830815428762736, −17.85307453285940552604205504030, −17.26947449095403173465870517673, −16.411442883943787453272511485002, −16.02962675456843167758160087709, −15.17936526949828614741426471508, −14.76630233182031698180136594888, −14.17384410161419555242459007187, −12.349772441900364373732716570407, −11.92831584826656692573819165981, −11.81377114051571099301077965287, −10.84325146899361719119390998752, −10.18868931043465330458540139474, −9.61755526454951193275295409976, −8.96632662564243236158607863482, −8.03187305952619984085730462646, −7.295186694573998453618173505366, −6.64193563141921866596318368226, −6.03791432402657410027233176734, −5.13455287241088403110401748887, −4.22945487602415283070673797763, −3.435781437095213030511556333788, −2.35827149988217128932782378136, −1.74690964204744056127364151704, −0.35842039148461165845825226774,
0.81530000234179214218619075503, 1.08341610815833454466266094452, 2.09051194191775156540229452998, 3.333067674116148537863633232, 4.179159530702522926641237262021, 5.134431290577634853907779109522, 5.84629061920123767561843997307, 6.63791795345850846588463413271, 7.45424706196552592207210311318, 7.936776130401295766300155389358, 8.43139829587139987750820679145, 9.48564814128207649076553910287, 10.16983234477863425985774458177, 11.07974635267208544239595532610, 11.31532350325988139765214987997, 12.07653311431662484081860108359, 12.81790183127458755461187781065, 13.36334395704859256243096193127, 14.49504679615018068425631144064, 15.22173157677426469280264515101, 16.198889863714037401305518460290, 16.51305143885561610580724331114, 17.21978746068200340177314102656, 17.530061647970811464168886085300, 18.276043177004871081369072162935