L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.988 − 0.149i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.900 + 0.433i)8-s + (0.955 + 0.294i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (0.365 − 0.930i)12-s + (0.0747 − 0.997i)13-s + (−0.988 + 0.149i)14-s + (−0.733 − 0.680i)15-s + (−0.900 − 0.433i)16-s + (0.826 − 0.563i)17-s + ⋯ |
L(s) = 1 | + (0.623 + 0.781i)2-s + (−0.988 − 0.149i)3-s + (−0.222 + 0.974i)4-s + (0.826 + 0.563i)5-s + (−0.5 − 0.866i)6-s + (−0.5 + 0.866i)7-s + (−0.900 + 0.433i)8-s + (0.955 + 0.294i)9-s + (0.0747 + 0.997i)10-s + (−0.222 − 0.974i)11-s + (0.365 − 0.930i)12-s + (0.0747 − 0.997i)13-s + (−0.988 + 0.149i)14-s + (−0.733 − 0.680i)15-s + (−0.900 − 0.433i)16-s + (0.826 − 0.563i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.0269 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6312039544 + 0.6144309601i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6312039544 + 0.6144309601i\) |
\(L(1)\) |
\(\approx\) |
\(0.8963836628 + 0.5413745073i\) |
\(L(1)\) |
\(\approx\) |
\(0.8963836628 + 0.5413745073i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 43 | \( 1 \) |
good | 2 | \( 1 + (0.623 + 0.781i)T \) |
| 3 | \( 1 + (-0.988 - 0.149i)T \) |
| 5 | \( 1 + (0.826 + 0.563i)T \) |
| 7 | \( 1 + (-0.5 + 0.866i)T \) |
| 11 | \( 1 + (-0.222 - 0.974i)T \) |
| 13 | \( 1 + (0.0747 - 0.997i)T \) |
| 17 | \( 1 + (0.826 - 0.563i)T \) |
| 19 | \( 1 + (0.955 - 0.294i)T \) |
| 23 | \( 1 + (-0.733 + 0.680i)T \) |
| 29 | \( 1 + (-0.988 + 0.149i)T \) |
| 31 | \( 1 + (0.365 - 0.930i)T \) |
| 37 | \( 1 + (-0.5 - 0.866i)T \) |
| 41 | \( 1 + (0.623 + 0.781i)T \) |
| 47 | \( 1 + (-0.222 + 0.974i)T \) |
| 53 | \( 1 + (0.0747 + 0.997i)T \) |
| 59 | \( 1 + (-0.900 - 0.433i)T \) |
| 61 | \( 1 + (0.365 + 0.930i)T \) |
| 67 | \( 1 + (0.955 - 0.294i)T \) |
| 71 | \( 1 + (-0.733 - 0.680i)T \) |
| 73 | \( 1 + (0.0747 - 0.997i)T \) |
| 79 | \( 1 + (-0.5 + 0.866i)T \) |
| 83 | \( 1 + (-0.988 - 0.149i)T \) |
| 89 | \( 1 + (-0.988 - 0.149i)T \) |
| 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
−33.83443255216844961805239608183, −33.07207425737266124533561085621, −32.26966289103927727674855613834, −30.591858199093243984543739763277, −29.4513173978352439234541552820, −28.72274853850344287010741858458, −27.89235228203764582884054022991, −26.2160416237976527768134634512, −24.37033135275627238924523445308, −23.35488766571014158893067784245, −22.38136425387917973928293862031, −21.20691775184445507511575705589, −20.275456532495087478952609483991, −18.59949840483354065370794629080, −17.28219229213822920255947176258, −16.10460503320502140744491174349, −14.19796169672790271091891479982, −12.94496537920759814447763876925, −11.98905908066448991388607816637, −10.34108839571527783695737858403, −9.672619435150973297878773662, −6.69329137984143319050504086237, −5.34041100712755420994037954811, −4.078543421435640068032895398476, −1.53587351594186967252160118865,
3.05843056762551056707845732391, 5.54015737798194062255898265238, 5.93239644544751770753859402002, 7.548762675684525216320346377912, 9.59790161289351084083442731052, 11.36076799283074629495149428170, 12.719761617234904940990205615637, 13.8191915697073123485313811102, 15.44909211982997308507344567467, 16.4653291768530653852702602739, 17.8074002397375762420920849133, 18.60167237013061575017267675416, 21.197198906423305052305517025536, 22.15928448097727831340765216564, 22.78271174028292973051466501783, 24.29734454020765955189420465767, 25.18604756030804506914996363736, 26.42281287977038442710131082158, 27.89290355951114601794328337008, 29.414392729542629154429544432383, 30.047304282845816050318641884270, 31.72438778648160853138277260000, 32.77477066640293680448594364133, 33.87733452247002868756759118861, 34.67057719693963064277728940225