L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)11-s + 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)28-s + (−0.5 + 0.866i)29-s − 31-s + (0.866 − 0.5i)32-s + ⋯ |
L(s) = 1 | + (−0.866 − 0.5i)2-s + (0.5 + 0.866i)4-s + (−0.866 + 0.5i)7-s − i·8-s + (−0.5 + 0.866i)11-s + 14-s + (−0.5 + 0.866i)16-s + (−0.866 + 0.5i)17-s + (−0.5 − 0.866i)19-s + (0.866 − 0.5i)22-s + (−0.866 − 0.5i)23-s + (−0.866 − 0.5i)28-s + (−0.5 + 0.866i)29-s − 31-s + (0.866 − 0.5i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 195 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.514 + 0.857i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1323416844 + 0.2338309119i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1323416844 + 0.2338309119i\) |
\(L(1)\) |
\(\approx\) |
\(0.4950080514 + 0.02789234401i\) |
\(L(1)\) |
\(\approx\) |
\(0.4950080514 + 0.02789234401i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-0.866 + 0.5i)T \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
| 17 | \( 1 + (-0.866 + 0.5i)T \) |
| 19 | \( 1 + (-0.5 - 0.866i)T \) |
| 23 | \( 1 + (-0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 + 0.866i)T \) |
| 31 | \( 1 - T \) |
| 37 | \( 1 + (0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.5 + 0.866i)T \) |
| 43 | \( 1 + (-0.866 + 0.5i)T \) |
| 47 | \( 1 + iT \) |
| 53 | \( 1 + iT \) |
| 59 | \( 1 + (0.5 + 0.866i)T \) |
| 61 | \( 1 + (-0.5 - 0.866i)T \) |
| 67 | \( 1 + (0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.5 - 0.866i)T \) |
| 73 | \( 1 + iT \) |
| 79 | \( 1 - T \) |
| 83 | \( 1 - iT \) |
| 89 | \( 1 + (0.5 - 0.866i)T \) |
| 97 | \( 1 + (-0.866 + 0.5i)T \) |
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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
−26.64119019765598805924426882732, −25.88088162885975573994117509338, −24.96955433531341411641622265363, −23.94996732410818193050611662084, −23.179464608266459782977535239130, −22.01902441379520569670357856749, −20.65799484452322663538762481479, −19.78415379611901379107365664237, −18.911845245428169912426016794943, −18.07528825600984806685571406876, −16.86348460866313604327877955419, −16.219542980265633764377301555724, −15.3384358754392334115684861731, −14.050488778466177403937778352200, −13.12700062726711515371911832925, −11.56932086550333371681003961559, −10.53744523630254507829582194236, −9.67468626930601423433179085613, −8.579459742303004074437783621244, −7.523107376841975966615094725217, −6.45139548585015341940340347578, −5.5043716862457026191766996427, −3.73479409657764535851107706286, −2.14707559297287447371045054684, −0.25347666483168864581514220129,
1.94275930450997271803684832681, 2.98278203540044493867625699944, 4.42987198654938389640293333837, 6.22759302995466086900592038286, 7.21880512951733363761281251621, 8.49053513260218190703221745410, 9.41149042416254533559496866297, 10.31938784791332340540003450379, 11.37261054208235866853865486430, 12.61724094753276970996342421400, 13.099699237042114364750958010068, 15.00152392943449888774035255238, 15.851612309154096323821574619172, 16.82088350800115125048710942994, 17.94217744519209753352796633095, 18.597321884871178871885611489131, 19.78978253174724235609568611606, 20.260742568109915468499201003119, 21.67731986929370620845449809383, 22.19729261651139091624803856884, 23.55990365298382697914826608765, 24.77861113651900600736639122745, 25.8606732362523333987629046489, 26.14669319514211693772822488761, 27.471839723214024596111329696238