L(s) = 1 | + (−0.415 − 0.909i)3-s + (0.142 + 0.989i)5-s + (−0.142 − 0.989i)7-s + (−0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (−0.415 − 0.909i)13-s + (0.841 − 0.540i)15-s + (0.841 + 0.540i)17-s + (0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)23-s + (−0.959 + 0.281i)25-s + (0.959 + 0.281i)27-s + (0.142 + 0.989i)29-s + (−0.654 − 0.755i)31-s + ⋯ |
L(s) = 1 | + (−0.415 − 0.909i)3-s + (0.142 + 0.989i)5-s + (−0.142 − 0.989i)7-s + (−0.654 + 0.755i)9-s + (0.142 − 0.989i)11-s + (−0.415 − 0.909i)13-s + (0.841 − 0.540i)15-s + (0.841 + 0.540i)17-s + (0.654 − 0.755i)19-s + (−0.841 + 0.540i)21-s + (−0.654 + 0.755i)23-s + (−0.959 + 0.281i)25-s + (0.959 + 0.281i)27-s + (0.142 + 0.989i)29-s + (−0.654 − 0.755i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 712 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.735 - 0.677i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.3061214273 - 0.7838918222i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3061214273 - 0.7838918222i\) |
\(L(1)\) |
\(\approx\) |
\(0.7644977835 - 0.3424993539i\) |
\(L(1)\) |
\(\approx\) |
\(0.7644977835 - 0.3424993539i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 89 | \( 1 \) |
good | 3 | \( 1 + (-0.415 - 0.909i)T \) |
| 5 | \( 1 + (0.142 + 0.989i)T \) |
| 7 | \( 1 + (-0.142 - 0.989i)T \) |
| 11 | \( 1 + (0.142 - 0.989i)T \) |
| 13 | \( 1 + (-0.415 - 0.909i)T \) |
| 17 | \( 1 + (0.841 + 0.540i)T \) |
| 19 | \( 1 + (0.654 - 0.755i)T \) |
| 23 | \( 1 + (-0.654 + 0.755i)T \) |
| 29 | \( 1 + (0.142 + 0.989i)T \) |
| 31 | \( 1 + (-0.654 - 0.755i)T \) |
| 37 | \( 1 - T \) |
| 41 | \( 1 + (0.415 - 0.909i)T \) |
| 43 | \( 1 + (0.142 - 0.989i)T \) |
| 47 | \( 1 + (0.415 - 0.909i)T \) |
| 53 | \( 1 + (-0.415 - 0.909i)T \) |
| 59 | \( 1 + (-0.415 + 0.909i)T \) |
| 61 | \( 1 + (0.959 + 0.281i)T \) |
| 67 | \( 1 + (-0.415 - 0.909i)T \) |
| 71 | \( 1 + (-0.142 + 0.989i)T \) |
| 73 | \( 1 + (-0.654 - 0.755i)T \) |
| 79 | \( 1 + (-0.654 - 0.755i)T \) |
| 83 | \( 1 + (-0.841 - 0.540i)T \) |
| 97 | \( 1 + (-0.142 - 0.989i)T \) |
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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
−22.78394638390826168805835434101, −22.04868718613300091649828868964, −21.20881707780474776612892062999, −20.70204533172306714364942527073, −19.89436698912370912552189047704, −18.79677791668253671956546401121, −17.8523974382016826424899943237, −17.07792098014533186611618800552, −16.18561100386453653562266726028, −15.87009536481834704571152720292, −14.703010141018072471197835721272, −14.10468802672826868236825048227, −12.52029367858402438219954217585, −12.192098638027180667327674034108, −11.45181977935000498944040514697, −9.96475953132363875282814544778, −9.599462019112869961869911788593, −8.843153492561135197896561135114, −7.79026092173350149311074516485, −6.4009680963864728447287511849, −5.522505229424141629375914877413, −4.807685893316917983164908661440, −4.03394593337118519532033655583, −2.689131287063336893705424403181, −1.462562512617850230492493343093,
0.432050564779851303834479259045, 1.67061890590116308806534929092, 2.983458360596731679828706048188, 3.67142642789551940688988158797, 5.39326877081081115009848074322, 5.971340964640255962350184735729, 7.19001038834123083453036420139, 7.40611774187555762780178521976, 8.537504040028689302352489180415, 9.99423233931317095321828170101, 10.669070654245982152779154478626, 11.36210190621124713062281471499, 12.312464256969725419121693351283, 13.3610688500842406731139106325, 13.88027264342410986172242336383, 14.627007880023406353933523559979, 15.8242172028034428971454602844, 16.80010792892633767509537187423, 17.5111043966861710313463005905, 18.146035534551030921503063946193, 19.10573214070350503994194494668, 19.57799945660395995198907995058, 20.498665387733658131179171077441, 21.874426308582232247984482282074, 22.27906226605728512650274639001