L(s) = 1 | − 2-s + (−0.809 − 0.587i)3-s + 4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (0.809 − 0.587i)7-s − 8-s + (0.309 + 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)15-s + 16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯ |
L(s) = 1 | − 2-s + (−0.809 − 0.587i)3-s + 4-s + (−0.809 − 0.587i)5-s + (0.809 + 0.587i)6-s + (0.809 − 0.587i)7-s − 8-s + (0.309 + 0.951i)9-s + (0.809 + 0.587i)10-s + (−0.809 − 0.587i)12-s + (0.309 − 0.951i)13-s + (−0.809 + 0.587i)14-s + (0.309 + 0.951i)15-s + 16-s + (0.809 + 0.587i)17-s + (−0.309 − 0.951i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 671 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.116 - 0.993i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5017712289 - 0.4464879153i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5017712289 - 0.4464879153i\) |
\(L(1)\) |
\(\approx\) |
\(0.5391093799 - 0.2182852907i\) |
\(L(1)\) |
\(\approx\) |
\(0.5391093799 - 0.2182852907i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 11 | \( 1 \) |
| 61 | \( 1 \) |
good | 2 | \( 1 - T \) |
| 3 | \( 1 + (-0.809 - 0.587i)T \) |
| 5 | \( 1 + (-0.809 - 0.587i)T \) |
| 7 | \( 1 + (0.809 - 0.587i)T \) |
| 13 | \( 1 + (0.309 - 0.951i)T \) |
| 17 | \( 1 + (0.809 + 0.587i)T \) |
| 19 | \( 1 + (-0.809 - 0.587i)T \) |
| 23 | \( 1 + (0.809 - 0.587i)T \) |
| 29 | \( 1 + (0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.809 + 0.587i)T \) |
| 37 | \( 1 + (0.809 - 0.587i)T \) |
| 41 | \( 1 + (0.309 + 0.951i)T \) |
| 43 | \( 1 + (0.809 + 0.587i)T \) |
| 47 | \( 1 + (-0.809 + 0.587i)T \) |
| 53 | \( 1 + (0.809 - 0.587i)T \) |
| 59 | \( 1 + (-0.309 + 0.951i)T \) |
| 67 | \( 1 + (0.809 - 0.587i)T \) |
| 71 | \( 1 + (-0.309 + 0.951i)T \) |
| 73 | \( 1 + (0.309 + 0.951i)T \) |
| 79 | \( 1 + (0.809 - 0.587i)T \) |
| 83 | \( 1 + (-0.809 + 0.587i)T \) |
| 89 | \( 1 + (-0.309 - 0.951i)T \) |
| 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
−23.17264883141195068253958579016, −21.98974491005231134419942019564, −21.10147197326052005180301743456, −20.76811351817066306436035673005, −19.33764960487981568692670027116, −18.76075838946989495191672765666, −18.115225288053884867602094318800, −17.17473672010215476949281243465, −16.50890163781554202058409814645, −15.585749063428267724792182425194, −15.11427406896788082584931091356, −14.21022571745639662690867502496, −12.30226621565551124683834651130, −11.723913205541556491968456826791, −11.20523155091321451281250632563, −10.399384226425185877498504329472, −9.47002254658836559990146343065, −8.53238317987460633660814933161, −7.651851531573702644093623147297, −6.68199701492653503911934984024, −5.86141148738403807666300232497, −4.66266201008462159537639448437, −3.59641527206842225395543109355, −2.36032023804033100490724494976, −0.96102990073132844601246343485,
0.76598890248952337960328386789, 1.31686008270889635657435825080, 2.836551651695105129419604515127, 4.34112010285018020694970336276, 5.30813185940473892805132704099, 6.42311815016276620173062162011, 7.35900648919412262072695534668, 8.081505595000969563929349668958, 8.59633858345270101578198204785, 10.135380968042789741724810559821, 10.90758466179873936557983117462, 11.39830593149238703175226324105, 12.462452085060873154587607410849, 12.94932110544509021383940603175, 14.491871531794724033397281394, 15.415308268700393539446319090795, 16.343950064186649703438717363563, 16.94240698435236163189562137393, 17.65932709799555393086870421318, 18.29969224251703872833381883283, 19.410807921871665337230479196513, 19.746509346666530676838350428254, 20.85278391369401571448075990975, 21.4683057347139295636632783375, 23.109752062029510204311611334224