L(s) = 1 | + (−0.984 + 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.342 + 0.939i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.342 − 0.939i)29-s + (−0.766 + 0.642i)31-s + (−0.866 + 0.5i)37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (−0.766 − 0.642i)47-s − i·53-s + 55-s + ⋯ |
L(s) = 1 | + (−0.984 + 0.173i)5-s + (−0.984 − 0.173i)11-s + (−0.342 + 0.939i)13-s + (0.5 + 0.866i)17-s + (0.866 + 0.5i)19-s + (−0.766 + 0.642i)23-s + (0.939 − 0.342i)25-s + (−0.342 − 0.939i)29-s + (−0.766 + 0.642i)31-s + (−0.866 + 0.5i)37-s + (−0.939 − 0.342i)41-s + (−0.984 − 0.173i)43-s + (−0.766 − 0.642i)47-s − i·53-s + 55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.435 - 0.900i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.1390667268 - 0.08718003235i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.1390667268 - 0.08718003235i\) |
\(L(1)\) |
\(\approx\) |
\(0.6619629611 + 0.1333241657i\) |
\(L(1)\) |
\(\approx\) |
\(0.6619629611 + 0.1333241657i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (-0.984 + 0.173i)T \) |
| 11 | \( 1 + (-0.984 - 0.173i)T \) |
| 13 | \( 1 + (-0.342 + 0.939i)T \) |
| 17 | \( 1 + (0.5 + 0.866i)T \) |
| 19 | \( 1 + (0.866 + 0.5i)T \) |
| 23 | \( 1 + (-0.766 + 0.642i)T \) |
| 29 | \( 1 + (-0.342 - 0.939i)T \) |
| 31 | \( 1 + (-0.766 + 0.642i)T \) |
| 37 | \( 1 + (-0.866 + 0.5i)T \) |
| 41 | \( 1 + (-0.939 - 0.342i)T \) |
| 43 | \( 1 + (-0.984 - 0.173i)T \) |
| 47 | \( 1 + (-0.766 - 0.642i)T \) |
| 53 | \( 1 - iT \) |
| 59 | \( 1 + (-0.984 + 0.173i)T \) |
| 61 | \( 1 + (0.642 - 0.766i)T \) |
| 67 | \( 1 + (0.342 - 0.939i)T \) |
| 71 | \( 1 + (0.5 + 0.866i)T \) |
| 73 | \( 1 + (-0.5 + 0.866i)T \) |
| 79 | \( 1 + (-0.939 + 0.342i)T \) |
| 83 | \( 1 + (0.342 + 0.939i)T \) |
| 89 | \( 1 + (-0.5 + 0.866i)T \) |
| 97 | \( 1 + (-0.173 + 0.984i)T \) |
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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
−18.902649181434761896682521847043, −18.30239018023247251150070641040, −17.823113981491556965259376270366, −16.69061106988324967998733552481, −16.16634398764757688433529632228, −15.55376965633569866177335439509, −14.94253423223176091089866102130, −14.15962588929949119337975440576, −13.23892915768308491583402362539, −12.61941042315430170085477831611, −11.97539363542168178150004181215, −11.260468394500344778322217402982, −10.48838416033613486772776629252, −9.822474554614170411674418518096, −8.89572489339800207830043701363, −8.084608284687007322419496425190, −7.5161290563273660118626065080, −7.00050961990099942786951684464, −5.68388478572666535503015430274, −5.098535110029338230279905147853, −4.421993670017722677799541831137, −3.22915572060077249347293300070, −2.93806020770684907350933951579, −1.64166631852505918699183330955, −0.43987030586424915803804208615,
0.05253194814946321977425835783, 1.402799073728722760332382774390, 2.26073620981644381113298550879, 3.49852710736560967524154953310, 3.71321943483761227438131200447, 4.910521794530126881458997833506, 5.47831563490767369792977540654, 6.56721711571109443391268694685, 7.25907289861353243359769271875, 8.05442883609705554015873125340, 8.41732571006211012221214505512, 9.61745473819656247294660615809, 10.1901441606514374580366340615, 11.02845865270778809199640391756, 11.77720881108522477751360683369, 12.20428204649229915275109872067, 13.109908710916291335528811817299, 13.902411088992446412520990570351, 14.58566164125607976971400862136, 15.364117295920944555222807699450, 15.908667120923162900864199054037, 16.569245658453666073581774355, 17.26034503102718272065615111912, 18.348470848950065691533111131752, 18.68738363264891143379763426021