L(s) = 1 | + (−0.971 − 0.235i)2-s + (−0.888 + 0.458i)3-s + (0.888 + 0.458i)4-s + (0.755 + 0.654i)5-s + (0.971 − 0.235i)6-s + (−0.189 + 0.981i)7-s + (−0.755 − 0.654i)8-s + (0.580 − 0.814i)9-s + (−0.580 − 0.814i)10-s + (0.189 + 0.981i)11-s − 12-s + (0.415 − 0.909i)14-s + (−0.971 − 0.235i)15-s + (0.580 + 0.814i)16-s + (−0.235 − 0.971i)17-s + (−0.755 + 0.654i)18-s + ⋯ |
L(s) = 1 | + (−0.971 − 0.235i)2-s + (−0.888 + 0.458i)3-s + (0.888 + 0.458i)4-s + (0.755 + 0.654i)5-s + (0.971 − 0.235i)6-s + (−0.189 + 0.981i)7-s + (−0.755 − 0.654i)8-s + (0.580 − 0.814i)9-s + (−0.580 − 0.814i)10-s + (0.189 + 0.981i)11-s − 12-s + (0.415 − 0.909i)14-s + (−0.971 − 0.235i)15-s + (0.580 + 0.814i)16-s + (−0.235 − 0.971i)17-s + (−0.755 + 0.654i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1157 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.596 + 0.802i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.9293637127 + 0.4674033631i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9293637127 + 0.4674033631i\) |
\(L(1)\) |
\(\approx\) |
\(0.6083499361 + 0.1675392802i\) |
\(L(1)\) |
\(\approx\) |
\(0.6083499361 + 0.1675392802i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 13 | \( 1 \) |
| 89 | \( 1 \) |
good | 2 | \( 1 + (-0.971 - 0.235i)T \) |
| 3 | \( 1 + (-0.888 + 0.458i)T \) |
| 5 | \( 1 + (0.755 + 0.654i)T \) |
| 7 | \( 1 + (-0.189 + 0.981i)T \) |
| 11 | \( 1 + (0.189 + 0.981i)T \) |
| 17 | \( 1 + (-0.235 - 0.971i)T \) |
| 19 | \( 1 + (-0.814 - 0.580i)T \) |
| 23 | \( 1 + (0.995 + 0.0950i)T \) |
| 29 | \( 1 + (-0.327 - 0.945i)T \) |
| 31 | \( 1 + (-0.909 - 0.415i)T \) |
| 37 | \( 1 + (-0.866 - 0.5i)T \) |
| 41 | \( 1 + (0.998 - 0.0475i)T \) |
| 43 | \( 1 + (0.327 - 0.945i)T \) |
| 47 | \( 1 + (0.540 + 0.841i)T \) |
| 53 | \( 1 + (0.841 + 0.540i)T \) |
| 59 | \( 1 + (0.458 - 0.888i)T \) |
| 61 | \( 1 + (0.928 - 0.371i)T \) |
| 67 | \( 1 + (0.458 + 0.888i)T \) |
| 71 | \( 1 + (0.945 + 0.327i)T \) |
| 73 | \( 1 + (0.909 + 0.415i)T \) |
| 79 | \( 1 + (0.415 - 0.909i)T \) |
| 83 | \( 1 + (-0.281 + 0.959i)T \) |
| 97 | \( 1 + (0.189 - 0.981i)T \) |
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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
−21.04637918934778096985938879211, −19.89860833444156650714174888517, −19.35763547675022039249688490261, −18.51978200466955548395160865705, −17.712550230406208883808478506894, −17.03104400017985507982211138225, −16.651054683862484745398432218381, −16.13834144478688687605574002809, −14.81901141959773329707029077740, −13.88137597499096318788047056391, −12.991730513144550632425592165944, −12.39662011692151454414948408072, −11.12726350272607701348701106345, −10.70683543188581689156505393469, −10.01072871203191568575597290601, −8.917937593649906576498857089231, −8.26662524069928807699326726344, −7.2012364126179947916485671226, −6.483625233374589915942134242187, −5.83082056269535126141181424743, −4.99241533712948013111824590852, −3.6711919237416900216527716461, −2.107915482014514427690905836261, −1.24516977860443469175583894087, −0.60410632188806741018046713222,
0.58268460244372295648058207523, 2.00939314795011508676483287247, 2.57641841018501453001433896178, 3.83678868621195109455503537719, 5.140021603004477519174187508743, 5.93618026777418500818382752467, 6.80095034089146284838323483265, 7.307843489538891021410528554423, 8.915292467467647579475163846191, 9.39480721501788601504023032939, 10.00974921145365609039749542729, 10.97227194878588699211021324174, 11.41302442421430567958414408600, 12.41192214631813643987429782745, 12.98885755353790394945560350023, 14.51682761697063777161358911769, 15.37296883135955546326478200541, 15.7180287068631561794239508903, 16.93477021327602252599340542080, 17.39674483885526254931612707108, 18.06602036744831802582008630445, 18.66066037650615116517689487857, 19.39965133380680578968679891351, 20.70797061920726750442561216424, 21.05297310818561716808596846625