L(s) = 1 | + (−0.123 − 0.992i)2-s + (−0.969 + 0.244i)4-s + (−0.964 − 0.263i)5-s + (−0.761 − 0.647i)7-s + (0.362 + 0.931i)8-s + (−0.142 + 0.989i)10-s + (−0.0665 + 0.997i)13-s + (−0.548 + 0.836i)14-s + (0.879 − 0.475i)16-s + (0.736 + 0.676i)17-s + (0.198 − 0.980i)19-s + (0.999 + 0.0190i)20-s + (−0.235 + 0.971i)23-s + (0.861 + 0.508i)25-s + (0.998 − 0.0570i)26-s + ⋯ |
L(s) = 1 | + (−0.123 − 0.992i)2-s + (−0.969 + 0.244i)4-s + (−0.964 − 0.263i)5-s + (−0.761 − 0.647i)7-s + (0.362 + 0.931i)8-s + (−0.142 + 0.989i)10-s + (−0.0665 + 0.997i)13-s + (−0.548 + 0.836i)14-s + (0.879 − 0.475i)16-s + (0.736 + 0.676i)17-s + (0.198 − 0.980i)19-s + (0.999 + 0.0190i)20-s + (−0.235 + 0.971i)23-s + (0.861 + 0.508i)25-s + (0.998 − 0.0570i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.291 + 0.956i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.1329265459 - 0.1793962702i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.1329265459 - 0.1793962702i\) |
\(L(1)\) |
\(\approx\) |
\(0.5397304670 - 0.3555831660i\) |
\(L(1)\) |
\(\approx\) |
\(0.5397304670 - 0.3555831660i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.123 - 0.992i)T \) |
| 5 | \( 1 + (-0.964 - 0.263i)T \) |
| 7 | \( 1 + (-0.761 - 0.647i)T \) |
| 13 | \( 1 + (-0.0665 + 0.997i)T \) |
| 17 | \( 1 + (0.736 + 0.676i)T \) |
| 19 | \( 1 + (0.198 - 0.980i)T \) |
| 23 | \( 1 + (-0.235 + 0.971i)T \) |
| 29 | \( 1 + (-0.00951 - 0.999i)T \) |
| 31 | \( 1 + (0.345 - 0.938i)T \) |
| 37 | \( 1 + (-0.466 - 0.884i)T \) |
| 41 | \( 1 + (0.290 - 0.956i)T \) |
| 43 | \( 1 + (0.0475 - 0.998i)T \) |
| 47 | \( 1 + (-0.820 + 0.572i)T \) |
| 53 | \( 1 + (0.0285 + 0.999i)T \) |
| 59 | \( 1 + (0.683 - 0.730i)T \) |
| 61 | \( 1 + (0.797 + 0.603i)T \) |
| 67 | \( 1 + (-0.327 - 0.945i)T \) |
| 71 | \( 1 + (-0.993 + 0.113i)T \) |
| 73 | \( 1 + (0.610 - 0.791i)T \) |
| 79 | \( 1 + (0.161 - 0.986i)T \) |
| 83 | \( 1 + (-0.997 - 0.0760i)T \) |
| 89 | \( 1 + (-0.415 + 0.909i)T \) |
| 97 | \( 1 + (0.964 - 0.263i)T \) |
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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
−22.26861809289211738521142236341, −21.07961111556378106415807886834, −19.95899982502500450876936421355, −19.33401457162331817086495087088, −18.368446699607207079734616076797, −18.221779237510480880564564580370, −16.84631067507254008436961809676, −16.093291789209244095478429564589, −15.80482525263789449062393433844, −14.7153490994727662729338020704, −14.42250151640259474601733149843, −13.05570054520252768251632806429, −12.4956234507829999303479959470, −11.66926805349284316656694828149, −10.31057673505845160361817111853, −9.848081344726373623119976420814, −8.58462355283685583482933721618, −8.16738428112492878530143437214, −7.197396862109917928987690987348, −6.48407748504302852692523799521, −5.53925880569804004507861226870, −4.75493061328714278742466333799, −3.53161415185340166319384991675, −2.95405681344647033137995641069, −1.02486017544695237107321789529,
0.07272602146292936998038288637, 0.893853806639546665318801594296, 2.11367462454801831790490688589, 3.34061965662633575319498170955, 3.92261343989039956697467381937, 4.63572770153585364355534344703, 5.86198393956496354342403159832, 7.17298983199609276559650919799, 7.802252033211572423370302475254, 8.880340724568202642997970675178, 9.531552882605722632879701256933, 10.39540992535553505018591600068, 11.28603376952270228066620681895, 11.88305460712322378305957789407, 12.66646793663768726572339941518, 13.44837012888082748673415980940, 14.14269708222650586695250086280, 15.265719480412176825572822361021, 16.12524577181843859107602571415, 16.946444440260563326683181625741, 17.55180646973274501451506330325, 18.87570093107167561756967453423, 19.253241118455870665017716774141, 19.7506528403896878571766172643, 20.63744175287166549617170293304