L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + 12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s − 23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (−0.809 − 0.587i)27-s + ⋯ |
L(s) = 1 | + (−0.809 + 0.587i)2-s + (0.309 + 0.951i)3-s + (0.309 − 0.951i)4-s + (−0.809 − 0.587i)6-s + (0.309 + 0.951i)8-s + (−0.809 + 0.587i)9-s + 12-s + (0.809 − 0.587i)13-s + (−0.809 − 0.587i)16-s + (0.809 + 0.587i)17-s + (0.309 − 0.951i)18-s + (0.309 + 0.951i)19-s − 23-s + (−0.809 + 0.587i)24-s + (−0.309 + 0.951i)26-s + (−0.809 − 0.587i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 385 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.605 + 0.795i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.4083598073 + 0.8237125192i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.4083598073 + 0.8237125192i\) |
\(L(1)\) |
\(\approx\) |
\(0.6599041163 + 0.4850398364i\) |
\(L(1)\) |
\(\approx\) |
\(0.6599041163 + 0.4850398364i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (0.309 + 0.951i)T \) |
| 13 | \( 1 + (0.809 - 0.587i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (0.309 + 0.951i)T \) |
| 23 | \( 1 - T \) |
| 29 | \( 1 + (-0.309 + 0.951i)T \) |
| 31 | \( 1 + (0.809 - 0.587i)T \) |
| 37 | \( 1 + (-0.309 + 0.951i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + T \) |
| 47 | \( 1 + (0.309 + 0.951i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 61 | \( 1 + (-0.809 - 0.587i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.809 - 0.587i)T \) |
| 73 | \( 1 + (-0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (0.809 + 0.587i)T \) |
| 89 | \( 1 - T \) |
| 97 | \( 1 + (-0.809 + 0.587i)T \) |
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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
−24.41937131415375673094527930689, −23.4030359750567613000005659735, −22.4393192157669017329788577759, −21.22197277243873843329453577445, −20.56964033858555178169359150171, −19.612163794578381885233307798843, −18.98387841784789964083391808397, −18.16082878057215838248427234903, −17.53546788989464440152798407926, −16.46737602613856668734791792515, −15.53635709099357449358022840816, −14.01544110284492381555363417121, −13.45117560776970288781810090302, −12.25150048466594068622158915872, −11.727093691421027092851235281291, −10.68571068056962627249812335744, −9.44190534862250200563407040906, −8.720252637004305841114266632210, −7.73146068561553044325461203729, −6.984441014902998757750248198686, −5.86765235189618134099112644476, −4.04039954075135206774990722128, −2.91218792223786168098219154995, −1.922888085920635106107230175618, −0.75178584337804524846713209451,
1.42204292788780681480340909884, 2.97704034293110944088297969613, 4.184181486530333075540767667215, 5.503581410707410954899337535097, 6.15139680602169060792624339704, 7.78097683795823430324897863524, 8.298573046549319451796722072959, 9.37004411149672321099563850217, 10.201615387360807480798028978806, 10.8182108184482719944016872455, 12.0115812363506588253110204394, 13.59798617652472531636567250923, 14.48541066814588587607501329516, 15.21388423142382840919830109891, 16.11498432795115069046388233628, 16.660217537824930462962943544935, 17.694916648877549816054612646200, 18.623024563522490798283999087779, 19.54787275460025494163979469418, 20.45395039574418069878908400480, 21.03909878344550352917136205669, 22.33257999907677308949202195787, 23.09467695387331780173742995196, 24.09908241089036441453300194497, 25.15493057302727861866549974420