L(s) = 1 | + (0.613 − 0.789i)2-s + (−0.775 − 0.631i)3-s + (−0.247 − 0.968i)4-s + (0.0227 − 0.999i)5-s + (−0.974 + 0.225i)6-s + (−0.998 + 0.0455i)7-s + (−0.917 − 0.398i)8-s + (0.203 + 0.979i)9-s + (−0.775 − 0.631i)10-s + (0.203 + 0.979i)11-s + (−0.419 + 0.907i)12-s + (−0.949 + 0.313i)13-s + (−0.576 + 0.816i)14-s + (−0.648 + 0.761i)15-s + (−0.877 + 0.480i)16-s + (−0.334 + 0.942i)17-s + ⋯ |
L(s) = 1 | + (0.613 − 0.789i)2-s + (−0.775 − 0.631i)3-s + (−0.247 − 0.968i)4-s + (0.0227 − 0.999i)5-s + (−0.974 + 0.225i)6-s + (−0.998 + 0.0455i)7-s + (−0.917 − 0.398i)8-s + (0.203 + 0.979i)9-s + (−0.775 − 0.631i)10-s + (0.203 + 0.979i)11-s + (−0.419 + 0.907i)12-s + (−0.949 + 0.313i)13-s + (−0.576 + 0.816i)14-s + (−0.648 + 0.761i)15-s + (−0.877 + 0.480i)16-s + (−0.334 + 0.942i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 967 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.957 - 0.287i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6749817411 - 0.09926390457i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6749817411 - 0.09926390457i\) |
\(L(1)\) |
\(\approx\) |
\(0.6701125716 - 0.4901770941i\) |
\(L(1)\) |
\(\approx\) |
\(0.6701125716 - 0.4901770941i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 967 | \( 1 \) |
good | 2 | \( 1 + (0.613 - 0.789i)T \) |
| 3 | \( 1 + (-0.775 - 0.631i)T \) |
| 5 | \( 1 + (0.0227 - 0.999i)T \) |
| 7 | \( 1 + (-0.998 + 0.0455i)T \) |
| 11 | \( 1 + (0.203 + 0.979i)T \) |
| 13 | \( 1 + (-0.949 + 0.313i)T \) |
| 17 | \( 1 + (-0.334 + 0.942i)T \) |
| 19 | \( 1 + (0.934 - 0.356i)T \) |
| 23 | \( 1 + (0.962 + 0.269i)T \) |
| 29 | \( 1 + (0.203 - 0.979i)T \) |
| 31 | \( 1 + (0.113 + 0.993i)T \) |
| 37 | \( 1 + (-0.949 + 0.313i)T \) |
| 41 | \( 1 + (0.682 + 0.730i)T \) |
| 43 | \( 1 + (0.983 + 0.181i)T \) |
| 47 | \( 1 + (-0.648 + 0.761i)T \) |
| 53 | \( 1 + (-0.715 + 0.699i)T \) |
| 59 | \( 1 + (-0.715 - 0.699i)T \) |
| 61 | \( 1 + (0.291 - 0.956i)T \) |
| 67 | \( 1 + (-0.917 + 0.398i)T \) |
| 71 | \( 1 + (-0.990 + 0.136i)T \) |
| 73 | \( 1 + (-0.715 - 0.699i)T \) |
| 79 | \( 1 + (-0.715 - 0.699i)T \) |
| 83 | \( 1 + (-0.974 + 0.225i)T \) |
| 89 | \( 1 + (0.803 + 0.595i)T \) |
| 97 | \( 1 + T \) |
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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
−22.15428106354429716629946405157, −21.42181222448011999824884603865, −20.489593680503430085997930180705, −19.28471252466852905442330811461, −18.42865246670289616198307644887, −17.65001418010896254696860134566, −16.82804346572654132356998996301, −16.13027533042245196408575886332, −15.61002174923847647185014738036, −14.709449928937074673965040083091, −14.0357931514101156543595247447, −13.105826017461892948463947158984, −12.146584874394903071952770214815, −11.45919295187952015850806092678, −10.54444014279504104181191849917, −9.62355341397247146251337671542, −8.88484768396499150848188352330, −7.30281138409118807409955863334, −6.96135933511845516161398555765, −5.94737774074632705648068794100, −5.43423895863888488282228850012, −4.305314134055673753648847805038, −3.260239612284701102165518713038, −2.88081547537163408648548968473, −0.30043652016104098840382438438,
1.071486983641055890150157903839, 1.92112057680695433311021229857, 3.01824919977894876491648023930, 4.41465232084294484499498403743, 4.88799377216216104777182965474, 5.87086749467364143702156368242, 6.65516264380169019220744753749, 7.59378181036705422904675625662, 9.06182481867915164329467755099, 9.68252690021880257728389594357, 10.50328808494818027752555740398, 11.63410368827234095196976872016, 12.23768395891506028398141154264, 12.79107438400814008733887539321, 13.28906267843723033358514411015, 14.285113285531806656112065087411, 15.475147273176941666736389223407, 16.111807281127565788985800036699, 17.31787189488907693890458192060, 17.56280425458478951423141343542, 18.9173927608056219065994714218, 19.49550787692213670668325900197, 19.96247071818377377558604457319, 20.999328649157675309220198904302, 21.85890581606005932470541657200