L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (0.994 − 0.104i)7-s + (0.587 − 0.809i)8-s + (0.5 − 0.866i)10-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)16-s + (0.978 − 0.207i)17-s + (0.994 + 0.104i)19-s + (0.207 − 0.978i)20-s + (0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (0.743 − 0.669i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
L(s) = 1 | + (0.951 − 0.309i)2-s + (0.809 − 0.587i)4-s + (0.743 − 0.669i)5-s + (0.994 − 0.104i)7-s + (0.587 − 0.809i)8-s + (0.5 − 0.866i)10-s + (0.913 − 0.406i)14-s + (0.309 − 0.951i)16-s + (0.978 − 0.207i)17-s + (0.994 + 0.104i)19-s + (0.207 − 0.978i)20-s + (0.5 + 0.866i)23-s + (0.104 − 0.994i)25-s + (0.743 − 0.669i)28-s + (−0.809 + 0.587i)29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1287 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.340 - 0.940i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(5.599861993 - 3.929484357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(5.599861993 - 3.929484357i\) |
\(L(1)\) |
\(\approx\) |
\(2.526332366 - 0.9989612244i\) |
\(L(1)\) |
\(\approx\) |
\(2.526332366 - 0.9989612244i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.951 - 0.309i)T \) |
| 5 | \( 1 + (0.743 - 0.669i)T \) |
| 7 | \( 1 + (0.994 - 0.104i)T \) |
| 17 | \( 1 + (0.978 - 0.207i)T \) |
| 19 | \( 1 + (0.994 + 0.104i)T \) |
| 23 | \( 1 + (0.5 + 0.866i)T \) |
| 29 | \( 1 + (-0.809 + 0.587i)T \) |
| 31 | \( 1 + (-0.207 + 0.978i)T \) |
| 37 | \( 1 + (0.994 - 0.104i)T \) |
| 41 | \( 1 + (0.994 + 0.104i)T \) |
| 43 | \( 1 + (0.5 - 0.866i)T \) |
| 47 | \( 1 + (-0.994 - 0.104i)T \) |
| 53 | \( 1 + (0.309 + 0.951i)T \) |
| 59 | \( 1 + (-0.587 - 0.809i)T \) |
| 61 | \( 1 + (0.669 + 0.743i)T \) |
| 67 | \( 1 + (-0.866 + 0.5i)T \) |
| 71 | \( 1 + (0.207 + 0.978i)T \) |
| 73 | \( 1 + (-0.587 - 0.809i)T \) |
| 79 | \( 1 + (0.669 - 0.743i)T \) |
| 83 | \( 1 + (0.207 + 0.978i)T \) |
| 89 | \( 1 + (-0.866 - 0.5i)T \) |
| 97 | \( 1 + (-0.207 + 0.978i)T \) |
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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
−20.917301598935321162000580708200, −20.77382432684366939280678248294, −19.5498671035167469211214857188, −18.490909694622784471267320755764, −17.896338986878639175222000351485, −17.00715329081078884507327986839, −16.4219923636940280580009101088, −15.25770237597823375354078037141, −14.644077973656415566838776610504, −14.23829985710145526348317062758, −13.369108676685627797185850543966, −12.64575944905884072684341010017, −11.54427487480890654693733567665, −11.15149162013252014953452910054, −10.17324295000170301457697778471, −9.215471801208008749699250193045, −7.909910275190701502345372648830, −7.510785621174396021698487371144, −6.39673869791179432193211010964, −5.70673098973372534910952882733, −5.00692358544268804446984825279, −4.02203328394840477629378430152, −2.97739278093878240479750344341, −2.23094589963293099677206302034, −1.222220785937229439222668316700,
1.05657077933494525061147023865, 1.50748332091096988362802271897, 2.62238304502357961752372294978, 3.635693725278515178300172957045, 4.6703892392294671233261063851, 5.36811189958064981892425927553, 5.80690625405959439879822522171, 7.13776908436017955170673339033, 7.811658278005516190805746318453, 9.04446042867470035228170514368, 9.79559580591007235967538406366, 10.69260725632505864311302252734, 11.48938171835630544605228073745, 12.21785491082571737178752603499, 12.97323608084679120490678788625, 13.80124240572388023435644342560, 14.30666009194528580149410877510, 15.036273395372241129186742039666, 16.11440774911645402715833765951, 16.66703336351593780990766926216, 17.64907823012411003170138825534, 18.341695123145089916557291792428, 19.38103175308074215524472575230, 20.31747981507470828590848190412, 20.711363621266196140953592538090