L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + i·6-s + 7-s − 8-s + (0.5 + 0.866i)9-s − i·11-s + (−0.866 + 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − i·17-s + (−0.5 + 0.866i)18-s + i·19-s + (0.866 + 0.5i)21-s + (0.866 − 0.5i)22-s + ⋯ |
L(s) = 1 | + (0.5 + 0.866i)2-s + (0.866 + 0.5i)3-s + (−0.5 + 0.866i)4-s + i·6-s + 7-s − 8-s + (0.5 + 0.866i)9-s − i·11-s + (−0.866 + 0.5i)12-s + (0.5 + 0.866i)14-s + (−0.5 − 0.866i)16-s − i·17-s + (−0.5 + 0.866i)18-s + i·19-s + (0.866 + 0.5i)21-s + (0.866 − 0.5i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2015 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.308 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(3.880613301 + 2.821274930i\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.880613301 + 2.821274930i\) |
\(L(1)\) |
\(\approx\) |
\(1.690510820 + 1.084919067i\) |
\(L(1)\) |
\(\approx\) |
\(1.690510820 + 1.084919067i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 13 | \( 1 \) |
| 31 | \( 1 \) |
good | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 3 | \( 1 + (0.866 + 0.5i)T \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 - iT \) |
| 17 | \( 1 - iT \) |
| 19 | \( 1 + iT \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
| 29 | \( 1 + (-0.5 - 0.866i)T \) |
| 37 | \( 1 + (0.5 - 0.866i)T \) |
| 41 | \( 1 - iT \) |
| 43 | \( 1 + iT \) |
| 47 | \( 1 + T \) |
| 53 | \( 1 + (0.866 + 0.5i)T \) |
| 59 | \( 1 - iT \) |
| 61 | \( 1 + (0.5 - 0.866i)T \) |
| 67 | \( 1 - T \) |
| 71 | \( 1 + (-0.866 + 0.5i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.5 - 0.866i)T \) |
| 83 | \( 1 + (0.5 - 0.866i)T \) |
| 89 | \( 1 + (0.866 - 0.5i)T \) |
| 97 | \( 1 + (0.5 - 0.866i)T \) |
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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
−19.69095781836727159778717175922, −19.16026393192397630357865815625, −18.15641140083745280349168403324, −17.86876371611550190705574558103, −16.955674530926593307360198665863, −15.312972877694392334591744504014, −15.05057190557390445126799133782, −14.504415197247463419716701870643, −13.50715525975456722366825318240, −13.126497795806474240007703105735, −12.27496682289633393830687230034, −11.64537009188173806273506973205, −10.76481084282055774032430392917, −10.02542241851349021825054073621, −9.06188299122455881449769687340, −8.61839397185170265637237400655, −7.53543772172231615313130730088, −6.871180183736812422130314792866, −5.732575219574921004596103365863, −4.768180829557648057008975276094, −4.168533470687922695439528932706, −3.19695545980431726096905154269, −2.33397346730958570825223668272, −1.64307037884539002141029892108, −0.94334931660651538192804251735,
0.69530810607707936683308630388, 2.14579032326667867018789684967, 3.0170588335290089013317883605, 3.85585177576793013500014318458, 4.5733841146036888207404828737, 5.33934877595089143915848247979, 6.09079456538675625253222227978, 7.41595399845411996737029011347, 7.71125056270805996514743187846, 8.70222357203770522732172860967, 9.001730196012355334045866569988, 10.11615259477399515289784750234, 11.09474662086477677689825381790, 11.797738504736830210674169710, 12.88635336699969199803173887444, 13.58724365245354287023771526984, 14.32104703000706996345731123561, 14.563541615815709715264953948716, 15.48391015053008826565667382730, 16.12409883190710485282703615949, 16.75786806180546216276873255532, 17.52248670521710504855692973832, 18.61211235221688318839137101388, 18.87043373525333796513744597559, 20.20825444699033471263277865112