L(s) = 1 | + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s + 7-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (−0.623 + 0.781i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + (0.222 − 0.974i)17-s + (0.623 + 0.781i)19-s + (−0.623 + 0.781i)20-s + (−0.900 + 0.433i)22-s + (−0.623 + 0.781i)23-s + ⋯ |
L(s) = 1 | + (0.900 + 0.433i)2-s + (0.623 + 0.781i)4-s + (0.222 + 0.974i)5-s + 7-s + (0.222 + 0.974i)8-s + (−0.222 + 0.974i)10-s + (−0.623 + 0.781i)11-s + (−0.222 − 0.974i)13-s + (0.900 + 0.433i)14-s + (−0.222 + 0.974i)16-s + (0.222 − 0.974i)17-s + (0.623 + 0.781i)19-s + (−0.623 + 0.781i)20-s + (−0.900 + 0.433i)22-s + (−0.623 + 0.781i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 129 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.113796740 + 2.561966540i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.113796740 + 2.561966540i\) |
\(L(1)\) |
\(\approx\) |
\(1.721633883 + 1.021940663i\) |
\(L(1)\) |
\(\approx\) |
\(1.721633883 + 1.021940663i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 43 | \( 1 \) |
good | 2 | \( 1 + (0.900 + 0.433i)T \) |
| 5 | \( 1 + (0.222 + 0.974i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + (-0.623 + 0.781i)T \) |
| 13 | \( 1 + (-0.222 - 0.974i)T \) |
| 17 | \( 1 + (0.222 - 0.974i)T \) |
| 19 | \( 1 + (0.623 + 0.781i)T \) |
| 23 | \( 1 + (-0.623 + 0.781i)T \) |
| 29 | \( 1 + (0.900 + 0.433i)T \) |
| 31 | \( 1 + (-0.900 - 0.433i)T \) |
| 37 | \( 1 + T \) |
| 41 | \( 1 + (0.900 + 0.433i)T \) |
| 47 | \( 1 + (-0.623 - 0.781i)T \) |
| 53 | \( 1 + (0.222 - 0.974i)T \) |
| 59 | \( 1 + (0.222 - 0.974i)T \) |
| 61 | \( 1 + (-0.900 + 0.433i)T \) |
| 67 | \( 1 + (0.623 + 0.781i)T \) |
| 71 | \( 1 + (-0.623 - 0.781i)T \) |
| 73 | \( 1 + (-0.222 - 0.974i)T \) |
| 79 | \( 1 + T \) |
| 83 | \( 1 + (0.900 - 0.433i)T \) |
| 89 | \( 1 + (0.900 - 0.433i)T \) |
| 97 | \( 1 + (0.623 - 0.781i)T \) |
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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
−28.54909886820976888887253211359, −27.54212772318265243118689717267, −26.191710151966587584213621180064, −24.692841386589433762496216116396, −24.10353638414998725104080981705, −23.4943671165981474273107088429, −21.75287060290344457434426440293, −21.35741496259107076728790961432, −20.38188921604248166354183060150, −19.40068861806402437042803538369, −18.102986983886232077217643012075, −16.72407521132864113927201332420, −15.79425413795140068263785024758, −14.45238462678235409919946548750, −13.66236855728114256378602144898, −12.56748286556636455643945268637, −11.593005062346316569596179388169, −10.5662350446761823954761552350, −9.12010917888080413705758630114, −7.84950007071385691497528917991, −6.12126516980477855881494782231, −5.04650140989337065279173087679, −4.156147115145855187476458714878, −2.348049872723634097666494786792, −1.060923281662657720335502926602,
2.11872618032725080037840825450, 3.31274841637385459497270162622, 4.87036973520268272171001600742, 5.81435662007521385345244337005, 7.350217373141528237637676981862, 7.88059648331419233844539963452, 9.96170499024211028184700430370, 11.12329456454616528678213312949, 12.11617365727199467841874216357, 13.42414219710417570588241685550, 14.44305377844456595228968559920, 15.07010144348501396576081250176, 16.18434635412521687587276235882, 17.75647662392822926812941559328, 18.13035859546211639032498661397, 20.04442428087432292910544995494, 20.907243539278450201776195922688, 21.91263790928665222688254809009, 22.84281218696032420015966155840, 23.5762391380506866841468130497, 24.8682727088595217500267684033, 25.51342193135585933269662257494, 26.648916157433042051874343507265, 27.5722939429326018506349135550, 29.22795111291217952248512659297