L(s) = 1 | + (−0.623 − 0.781i)2-s + (0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.900 − 0.433i)11-s − 12-s + (0.900 + 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s − 17-s + (0.900 + 0.433i)18-s + (−0.222 + 0.974i)19-s + ⋯ |
L(s) = 1 | + (−0.623 − 0.781i)2-s + (0.222 + 0.974i)3-s + (−0.222 + 0.974i)4-s + (0.623 − 0.781i)6-s + (0.222 + 0.974i)7-s + (0.900 − 0.433i)8-s + (−0.900 + 0.433i)9-s + (−0.900 − 0.433i)11-s − 12-s + (0.900 + 0.433i)13-s + (0.623 − 0.781i)14-s + (−0.900 − 0.433i)16-s − 17-s + (0.900 + 0.433i)18-s + (−0.222 + 0.974i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 145 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.128 + 0.991i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5283904289 + 0.4643430932i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5283904289 + 0.4643430932i\) |
\(L(1)\) |
\(\approx\) |
\(0.7268306893 + 0.1919772141i\) |
\(L(1)\) |
\(\approx\) |
\(0.7268306893 + 0.1919772141i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 29 | \( 1 \) |
good | 2 | \( 1 + (-0.623 - 0.781i)T \) |
| 3 | \( 1 + (0.222 + 0.974i)T \) |
| 7 | \( 1 + (0.222 + 0.974i)T \) |
| 11 | \( 1 + (-0.900 - 0.433i)T \) |
| 13 | \( 1 + (0.900 + 0.433i)T \) |
| 17 | \( 1 - T \) |
| 19 | \( 1 + (-0.222 + 0.974i)T \) |
| 23 | \( 1 + (-0.623 + 0.781i)T \) |
| 31 | \( 1 + (0.623 + 0.781i)T \) |
| 37 | \( 1 + (0.900 - 0.433i)T \) |
| 41 | \( 1 + T \) |
| 43 | \( 1 + (-0.623 + 0.781i)T \) |
| 47 | \( 1 + (0.900 + 0.433i)T \) |
| 53 | \( 1 + (-0.623 - 0.781i)T \) |
| 59 | \( 1 + T \) |
| 61 | \( 1 + (-0.222 - 0.974i)T \) |
| 67 | \( 1 + (0.900 - 0.433i)T \) |
| 71 | \( 1 + (-0.900 - 0.433i)T \) |
| 73 | \( 1 + (-0.623 + 0.781i)T \) |
| 79 | \( 1 + (-0.900 + 0.433i)T \) |
| 83 | \( 1 + (0.222 - 0.974i)T \) |
| 89 | \( 1 + (0.623 + 0.781i)T \) |
| 97 | \( 1 + (0.222 - 0.974i)T \) |
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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
−28.10799504719426432334743228843, −26.60240972174548646349696696181, −26.09409459014991835405680126960, −25.12760092461751341973584153350, −24.02354047871584940059065994142, −23.55920837294733081113129191131, −22.58348468867687232991770705393, −20.507309031549710780993319182741, −19.96420074290775081490229176870, −18.72322998141409529306957470975, −17.90205294534059457977177205, −17.242284793635746873965478240702, −15.920286473433190602561875055893, −14.87866075398225969026054088714, −13.63303551021645571475174289687, −13.11973547553402707761095596758, −11.25935957408459739774683376576, −10.30052517157561856724024957642, −8.78647490167904093138411490100, −7.89196599556513121706416900895, −7.01683826362336735252348391887, −6.00587702115568388921559071386, −4.4621661529749354543305285602, −2.31826943645423395669771680452, −0.73475961966970179134532702205,
2.079345067882472119038615344582, 3.25160225345559796985255635189, 4.48808589866343090969546155527, 5.910880354541151273323698258878, 8.049970610407453929256567220191, 8.74187932240672495719230073270, 9.74883212964453398665736168421, 10.87654800560407029613546896049, 11.59115603357842104286380182114, 12.97495966900626017903822437139, 14.1757625104475267751557338466, 15.68742395674225610508936132819, 16.19711224887297133206486824542, 17.630185987601800929801607661226, 18.534353068393804533749965175904, 19.54011454260497885016857382042, 20.70982795912479672501615771994, 21.34533627839123870266902901850, 22.033492313695805910303424475441, 23.26190761373841646574935132524, 24.93359669600011781567019450236, 25.872227269064260744935882389467, 26.65320778795882874762843934886, 27.562821465258759011150674739, 28.42993690434983436054601065125