L(s) = 1 | + (−0.906 + 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s + 17-s + (0.707 + 0.707i)19-s + (−0.984 + 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (−0.766 + 0.642i)31-s + (−0.258 − 0.965i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
L(s) = 1 | + (−0.906 + 0.422i)5-s + (−0.422 + 0.906i)11-s + (0.573 − 0.819i)13-s + 17-s + (0.707 + 0.707i)19-s + (−0.984 + 0.173i)23-s + (0.642 − 0.766i)25-s + (−0.819 + 0.573i)29-s + (−0.766 + 0.642i)31-s + (−0.258 − 0.965i)37-s + (0.984 − 0.173i)41-s + (−0.996 + 0.0871i)43-s + (−0.766 − 0.642i)47-s + (−0.258 − 0.965i)53-s − i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 6048 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.811 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01262841798 - 0.03919834855i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01262841798 - 0.03919834855i\) |
\(L(1)\) |
\(\approx\) |
\(0.7757387298 + 0.09896829270i\) |
\(L(1)\) |
\(\approx\) |
\(0.7757387298 + 0.09896829270i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.906 + 0.422i)T \) |
| 11 | \( 1 + (-0.422 + 0.906i)T \) |
| 13 | \( 1 + (0.573 - 0.819i)T \) |
| 17 | \( 1 + T \) |
| 19 | \( 1 + (0.707 + 0.707i)T \) |
| 23 | \( 1 + (-0.984 + 0.173i)T \) |
| 29 | \( 1 + (-0.819 + 0.573i)T \) |
| 31 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 + (-0.258 - 0.965i)T \) |
| 41 | \( 1 + (0.984 - 0.173i)T \) |
| 43 | \( 1 + (-0.996 + 0.0871i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 + (-0.258 - 0.965i)T \) |
| 59 | \( 1 + (-0.819 - 0.573i)T \) |
| 61 | \( 1 + (0.0871 + 0.996i)T \) |
| 67 | \( 1 + (0.422 + 0.906i)T \) |
| 71 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 + (0.866 - 0.5i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (-0.819 + 0.573i)T \) |
| 89 | \( 1 - iT \) |
| 97 | \( 1 + (0.766 + 0.642i)T \) |
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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
−18.22462528497777109759547914112, −17.005778235151382608101239247125, −16.660278241974016978981079760958, −15.92719136305962365011178225616, −15.61301119625340115285343495494, −14.67785461721652632922811395908, −14.01119340699735553859109587886, −13.375384071312411006554403674717, −12.720626961741846234610008712794, −11.87502720639309482082573319463, −11.44119244051574955329102112028, −10.918062346217726795990153855159, −9.938551470658000401977634582777, −9.22925188060358024424355791461, −8.60379254470318357735209437523, −7.80397000532028971326827304429, −7.53235118302592640770231102800, −6.41168191672813326003329893061, −5.80516662321008098902867165690, −4.99204092926772750264193653344, −4.26748622207526859823632009314, −3.5190638593413810560175892591, −2.99566251323859272705567313080, −1.80668342609042458783690581281, −0.97773917557450007347852342999,
0.0120559332483328627232480064, 1.24587773736553381965721487579, 2.0788521059505129777887350385, 3.19050163427112355617700557198, 3.547560082150380257205166218427, 4.323629796058544453744511067729, 5.35810760907143944965272656044, 5.72701577628479309544804903626, 6.87534060666379234304774597675, 7.43088945176822699695321348708, 7.96480177822497432252254100940, 8.55452494743707325218808468303, 9.66710969949845490261954465008, 10.13391308845077563669974797942, 10.848118754969326487738420008314, 11.474056504328888717695576012421, 12.3472284237641622457168952463, 12.56955398763658931474694680768, 13.48919570176923682598719685992, 14.48633406384071888802159844730, 14.67944495639761747812544862588, 15.56001924986504011682472963621, 16.0997413965999625019172914644, 16.53968311157736505086059826092, 17.66544819507593335414813990008