L(s) = 1 | + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)10-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s − 20-s + (0.173 − 0.984i)22-s + (−0.766 + 0.642i)23-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯ |
L(s) = 1 | + (0.939 − 0.342i)2-s + (0.766 − 0.642i)4-s + (−0.766 − 0.642i)5-s + (−0.5 − 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.939 − 0.342i)10-s + (0.5 − 0.866i)11-s + (0.173 + 0.984i)13-s + (−0.766 − 0.642i)14-s + (0.173 − 0.984i)16-s + (0.939 − 0.342i)17-s − 20-s + (0.173 − 0.984i)22-s + (−0.766 + 0.642i)23-s + (0.173 + 0.984i)25-s + (0.5 + 0.866i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 57 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.158 - 0.987i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.487603150 - 1.744667923i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.487603150 - 1.744667923i\) |
\(L(1)\) |
\(\approx\) |
\(1.432338725 - 0.7970462988i\) |
\(L(1)\) |
\(\approx\) |
\(1.432338725 - 0.7970462988i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 19 | \( 1 \) |
good | 2 | \( 1 + (0.939 - 0.342i)T \) |
| 5 | \( 1 + (-0.766 - 0.642i)T \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 11 | \( 1 + (0.5 - 0.866i)T \) |
| 13 | \( 1 + (0.173 + 0.984i)T \) |
| 17 | \( 1 + (0.939 - 0.342i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (0.939 + 0.342i)T \) |
| 31 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (-0.173 + 0.984i)T \) |
| 43 | \( 1 + (0.766 + 0.642i)T \) |
| 47 | \( 1 + (0.939 + 0.342i)T \) |
| 53 | \( 1 + (-0.766 + 0.642i)T \) |
| 59 | \( 1 + (0.939 - 0.342i)T \) |
| 61 | \( 1 + (0.766 - 0.642i)T \) |
| 67 | \( 1 + (-0.939 - 0.342i)T \) |
| 71 | \( 1 + (-0.766 - 0.642i)T \) |
| 73 | \( 1 + (0.173 - 0.984i)T \) |
| 79 | \( 1 + (0.173 - 0.984i)T \) |
| 83 | \( 1 + (0.5 + 0.866i)T \) |
| 89 | \( 1 + (-0.173 - 0.984i)T \) |
| 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
−32.72351741331756084692814852070, −31.90347454065590589624427307831, −30.71055795379032466269711384273, −30.137236741982695865441997673305, −28.57156754324906678876318055774, −27.29862724930433629416499058668, −25.76709043411188836528727696592, −25.126560719700477676177448943145, −23.62775831544872019236406510771, −22.68472787282159763879324566920, −21.973427745839588717970703241394, −20.43999517968585527682271201544, −19.29797257377328065230237966471, −17.82003742197605010744012566835, −16.15854235941930140733366736820, −15.24854935876453587352929594401, −14.38448227990265936168596498852, −12.608991274649940151534949771049, −11.9363996521862985190752233335, −10.34996817506058062523931927079, −8.262335110489794748759976652797, −6.9647668272336937646989893728, −5.685943132377520305179438358776, −3.96161010901750841529312158243, −2.6708859450365844379171057466,
1.02652290762661720039895217833, 3.43586172547184364963336872634, 4.42903168576556116253493824770, 6.14580844086326038786558491921, 7.61334218502991867023541367650, 9.51983830615276684184325291224, 11.1312029374982435726857893126, 12.09285028396950352242371669084, 13.39348842337175135584333154712, 14.384008030660038420994831631998, 16.05603381664387255845376518415, 16.627034487874237354771434520054, 19.02311024288493592671242525301, 19.80855512113806804760314718792, 20.87431964141885983516419997389, 22.09628043401327683107690731479, 23.45312389681413586728279714737, 23.89325505950998263047796475042, 25.28971718656723079210779019360, 26.846100467207790934130025006603, 28.06689896494682738041684780111, 29.24371174985357504416179301991, 30.126998069486563292680013303659, 31.41615456825326222497843099063, 32.18046202254754291486752433763