L(s) = 1 | + (−0.207 + 0.978i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (0.951 + 0.309i)6-s + (0.587 − 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.587 + 0.809i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (0.406 − 0.913i)18-s + (0.406 + 0.913i)19-s + (0.951 − 0.309i)20-s + ⋯ |
L(s) = 1 | + (−0.207 + 0.978i)2-s + (0.104 − 0.994i)3-s + (−0.913 − 0.406i)4-s + (−0.743 + 0.669i)5-s + (0.951 + 0.309i)6-s + (0.587 − 0.809i)8-s + (−0.978 − 0.207i)9-s + (−0.5 − 0.866i)10-s + (−0.5 + 0.866i)12-s + (0.587 + 0.809i)15-s + (0.669 + 0.743i)16-s + (−0.978 + 0.207i)17-s + (0.406 − 0.913i)18-s + (0.406 + 0.913i)19-s + (0.951 − 0.309i)20-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1001 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.608 - 0.793i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.5781255976 - 0.2852916786i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.5781255976 - 0.2852916786i\) |
\(L(1)\) |
\(\approx\) |
\(0.6919623459 + 0.08692173660i\) |
\(L(1)\) |
\(\approx\) |
\(0.6919623459 + 0.08692173660i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (-0.207 + 0.978i)T \) |
| 3 | \( 1 + (0.104 - 0.994i)T \) |
| 5 | \( 1 + (-0.743 + 0.669i)T \) |
| 17 | \( 1 + (-0.978 + 0.207i)T \) |
| 19 | \( 1 + (0.406 + 0.913i)T \) |
| 23 | \( 1 + (0.5 - 0.866i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (0.743 + 0.669i)T \) |
| 37 | \( 1 + (0.994 - 0.104i)T \) |
| 41 | \( 1 + (0.587 - 0.809i)T \) |
| 43 | \( 1 - T \) |
| 47 | \( 1 + (-0.406 - 0.913i)T \) |
| 53 | \( 1 + (0.669 - 0.743i)T \) |
| 59 | \( 1 + (0.406 - 0.913i)T \) |
| 61 | \( 1 + (-0.669 - 0.743i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (-0.951 - 0.309i)T \) |
| 73 | \( 1 + (0.406 - 0.913i)T \) |
| 79 | \( 1 + (-0.978 - 0.207i)T \) |
| 83 | \( 1 + (0.951 + 0.309i)T \) |
| 89 | \( 1 + (0.866 + 0.5i)T \) |
| 97 | \( 1 + (-0.951 + 0.309i)T \) |
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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
−21.52807433012807787872946849797, −20.97869972012103298691257867564, −20.01268291325215480978408566590, −19.878445015979007504795624181777, −18.92749876893053160425830537268, −17.80991225000084128995147522985, −17.087131841764001989697103581722, −16.338381719418795599527806381679, −15.463053464598020524047124043696, −14.81534534465908566308341520472, −13.48198111016390946176973401863, −13.13390991162042782642571752953, −11.67523692985655199523459696116, −11.52898493377456053218163511624, −10.635728282416480464000512406970, −9.499316896609183270855481634716, −9.166510731059885637381184639993, −8.2695252634351234561839391360, −7.44213371599198040425284929558, −5.75508645068077674162944696212, −4.67620862355103226507307639115, −4.34094839351802357746797537496, −3.310216266014912567137152949541, −2.47916275624844273884183774243, −0.9880008695382722966320440265,
0.36586091985398793879876980475, 1.754185603770180501319575481476, 3.06027722160328359363648517736, 4.06110311387649889089459749684, 5.21091395317073958100256232106, 6.3098828740726710686767831700, 6.81833751292084238005126993707, 7.59909057008385221366500403100, 8.29489008567730610618696304621, 9.00609288314129692685535405802, 10.262351874264216508393309620688, 11.12845095531807327280278834300, 12.11005760818456753414236960509, 12.95043549105462556587551208504, 13.75195676926077034723652111629, 14.61930090275358972914128178359, 15.0271942696064612590860102699, 16.10167478259468829931750285796, 16.79592050868170205933741291804, 17.84142274605604649696552299290, 18.295408168927402533925536199705, 19.01556411945475875348915493427, 19.628773915889133310875415000783, 20.47891552672671706168976572999, 21.95925383489325709965082806578