L(s) = 1 | + (0.865 − 0.500i)2-s + (−0.134 + 0.990i)3-s + (0.498 − 0.867i)4-s + (−0.949 − 0.312i)5-s + (0.380 + 0.924i)6-s + (0.999 − 0.0138i)7-s + (−0.00345 − 0.999i)8-s + (−0.963 − 0.266i)9-s + (−0.978 + 0.205i)10-s + (0.792 + 0.609i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.436 − 0.899i)15-s + (−0.503 − 0.863i)16-s + (−0.847 − 0.530i)17-s + (−0.967 + 0.252i)18-s + ⋯ |
L(s) = 1 | + (0.865 − 0.500i)2-s + (−0.134 + 0.990i)3-s + (0.498 − 0.867i)4-s + (−0.949 − 0.312i)5-s + (0.380 + 0.924i)6-s + (0.999 − 0.0138i)7-s + (−0.00345 − 0.999i)8-s + (−0.963 − 0.266i)9-s + (−0.978 + 0.205i)10-s + (0.792 + 0.609i)12-s + (−0.753 − 0.657i)13-s + (0.858 − 0.512i)14-s + (0.436 − 0.899i)15-s + (−0.503 − 0.863i)16-s + (−0.847 − 0.530i)17-s + (−0.967 + 0.252i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6017 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.457 + 0.889i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1093862759 + 0.1793096183i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1093862759 + 0.1793096183i\) |
\(L(1)\) |
\(\approx\) |
\(1.121445506 - 0.2406331864i\) |
\(L(1)\) |
\(\approx\) |
\(1.121445506 - 0.2406331864i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 547 | \( 1 \) |
good | 2 | \( 1 + (0.865 - 0.500i)T \) |
| 3 | \( 1 + (-0.134 + 0.990i)T \) |
| 5 | \( 1 + (-0.949 - 0.312i)T \) |
| 7 | \( 1 + (0.999 - 0.0138i)T \) |
| 13 | \( 1 + (-0.753 - 0.657i)T \) |
| 17 | \( 1 + (-0.847 - 0.530i)T \) |
| 19 | \( 1 + (-0.998 + 0.0552i)T \) |
| 23 | \( 1 + (0.479 - 0.877i)T \) |
| 29 | \( 1 + (-0.965 - 0.259i)T \) |
| 31 | \( 1 + (-0.788 + 0.615i)T \) |
| 37 | \( 1 + (0.467 - 0.883i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (0.770 - 0.636i)T \) |
| 47 | \( 1 + (-0.861 - 0.506i)T \) |
| 53 | \( 1 + (-0.882 + 0.470i)T \) |
| 59 | \( 1 + (0.168 + 0.985i)T \) |
| 61 | \( 1 + (0.986 - 0.164i)T \) |
| 67 | \( 1 + (-0.792 - 0.609i)T \) |
| 71 | \( 1 + (-0.424 + 0.905i)T \) |
| 73 | \( 1 + (-0.175 + 0.984i)T \) |
| 79 | \( 1 + (-0.249 + 0.968i)T \) |
| 83 | \( 1 + (0.836 - 0.548i)T \) |
| 89 | \( 1 + (0.700 + 0.713i)T \) |
| 97 | \( 1 + (0.969 - 0.246i)T \) |
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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
−17.52919772152004759184498589885, −16.8336081895832129768394493678, −16.2100050711048674467327589421, −15.19335550200396744301630298655, −14.79758133154910968797301799770, −14.41336344811591116387103174781, −13.52319789515892622102242640545, −12.88641462008625464688218184958, −12.38013451881442744247133753636, −11.480003804071092056130123559038, −11.34744099966334017235145996865, −10.715927475080945209951026851797, −9.12463275251308280767036104753, −8.51445244643812207610075391262, −7.690560458065527718687364400593, −7.51100383447052445914626816843, −6.7110350322905840184037335271, −6.13633480964097762723544063266, −5.20320823727951546895951897292, −4.56749682400953162395251218091, −3.9313525632791182975792315800, −3.014582567825526044710090278420, −2.134921645178015048226212064560, −1.65568827672159803594866022069, −0.04057908698462911344709910891,
0.94773087065985805783512693951, 2.222156223521370445728682354993, 2.768292364334633742363933693471, 3.81963207201993221573089854096, 4.20707483243086122978279360719, 4.95218384207230743919751718385, 5.20730514731335925666230104069, 6.203678054637539804562737977283, 7.15317077666261226605011128072, 7.86720919714054250720755636987, 8.77010062817456909096856003636, 9.27025280904879553048267835624, 10.32518154293006471099785996451, 10.84368296164066943073940467374, 11.28852539762083205826061371104, 11.866380224657407359862465852347, 12.63513236512365126384648268034, 13.14791701804687083591016412807, 14.32737084936738622320727943028, 14.7249179912050705257263564144, 15.09842505017861331701747237100, 15.79568307783595864094784064999, 16.40818010721024056959971079877, 17.10407793092492847875468213758, 17.885294296984842433191611040019