L(s) = 1 | + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (−0.642 − 0.766i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (−0.766 + 0.642i)31-s − i·37-s + (0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.766 − 0.642i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯ |
L(s) = 1 | + (0.642 + 0.766i)5-s + (0.642 − 0.766i)11-s + (−0.642 − 0.766i)13-s + (0.5 − 0.866i)17-s + (−0.866 + 0.5i)19-s + (0.939 + 0.342i)23-s + (−0.173 + 0.984i)25-s + (−0.642 + 0.766i)29-s + (−0.766 + 0.642i)31-s − i·37-s + (0.766 − 0.642i)41-s + (0.342 + 0.939i)43-s + (−0.766 − 0.642i)47-s + (−0.866 + 0.5i)53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.452 - 0.891i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6262984375 - 1.020114947i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6262984375 - 1.020114947i\) |
\(L(1)\) |
\(\approx\) |
\(1.089733430 + 0.01340832104i\) |
\(L(1)\) |
\(\approx\) |
\(1.089733430 + 0.01340832104i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (0.642 + 0.766i)T \) |
| 11 | \( 1 + (0.642 - 0.766i)T \) |
| 13 | \( 1 + (-0.642 - 0.766i)T \) |
| 17 | \( 1 + (0.5 - 0.866i)T \) |
| 19 | \( 1 + (-0.866 + 0.5i)T \) |
| 23 | \( 1 + (0.939 + 0.342i)T \) |
| 29 | \( 1 + (-0.642 + 0.766i)T \) |
| 31 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 - iT \) |
| 41 | \( 1 + (0.766 - 0.642i)T \) |
| 43 | \( 1 + (0.342 + 0.939i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.866 + 0.5i)T \) |
| 59 | \( 1 + (-0.984 + 0.173i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.642 - 0.766i)T \) |
| 89 | \( 1 + (-0.5 - 0.866i)T \) |
| 97 | \( 1 + (0.939 - 0.342i)T \) |
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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
−19.2485577100990180734386477811, −18.40810108169391081377115432987, −17.3983721256863129300341579807, −17.036191004752830659294522589087, −16.65141829404345872641142938134, −15.55455308214831360405294604123, −14.75139310681473721651231159354, −14.37574644135345266807983120587, −13.27949280081515631751406435681, −12.82872803643055240182273251178, −12.173623550486094177521770613923, −11.3836090701071772450151303672, −10.506830720947329481123990228011, −9.57972799493992920107029669353, −9.31172015775424446597081536095, −8.44286881809131981178266667090, −7.60550517449541465915962637189, −6.66602730159410819673558283076, −6.12689338031941664904392721568, −5.10625601226404737944952120229, −4.50439469938339646298554847102, −3.80425596489335720894043931907, −2.46168371578832504122546963121, −1.85198552407524291447188098819, −1.0196225002132857896482812052,
0.187003628354423961271200738025, 1.2805665451953778359500671120, 2.20787961199914330712649680169, 3.13021709798463526648896932917, 3.59182346663348529649029601333, 4.89241463634142895913570846507, 5.59240241213523311803490394827, 6.25218885514924476994599212674, 7.125867857439776601710722741707, 7.65259647597685719275934332681, 8.77004057425089834836684074526, 9.39097794221764086232227044217, 10.09746092028198467162229227716, 10.9782258854020701694555908157, 11.26148560420428947672225941962, 12.519200697306936391279573685634, 12.88201953788515552446803147290, 14.01753454932270345242615106677, 14.34422678216629644653502170175, 14.96922157751768029938145200537, 15.86009503176638750237691291353, 16.73842217376081992251602466506, 17.20145900383383625199753892234, 18.02208826561129735266291594270, 18.60275449107347614343597669954