L(s) = 1 | + (0.405 − 0.914i)2-s + (−0.0174 − 0.999i)3-s + (−0.671 − 0.740i)4-s + (0.961 + 0.276i)5-s + (−0.921 − 0.389i)6-s + (0.472 − 0.881i)7-s + (−0.949 + 0.314i)8-s + (−0.999 + 0.0349i)9-s + (0.641 − 0.766i)10-s + (−0.598 − 0.801i)11-s + (−0.728 + 0.684i)12-s + (−0.418 + 0.908i)13-s + (−0.614 − 0.788i)14-s + (0.259 − 0.965i)15-s + (−0.0972 + 0.995i)16-s + (−0.902 + 0.430i)17-s + ⋯ |
L(s) = 1 | + (0.405 − 0.914i)2-s + (−0.0174 − 0.999i)3-s + (−0.671 − 0.740i)4-s + (0.961 + 0.276i)5-s + (−0.921 − 0.389i)6-s + (0.472 − 0.881i)7-s + (−0.949 + 0.314i)8-s + (−0.999 + 0.0349i)9-s + (0.641 − 0.766i)10-s + (−0.598 − 0.801i)11-s + (−0.728 + 0.684i)12-s + (−0.418 + 0.908i)13-s + (−0.614 − 0.788i)14-s + (0.259 − 0.965i)15-s + (−0.0972 + 0.995i)16-s + (−0.902 + 0.430i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1259 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.799 - 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.483256385 - 0.4945005614i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.483256385 - 0.4945005614i\) |
\(L(1)\) |
\(\approx\) |
\(0.8670174799 - 0.8154241131i\) |
\(L(1)\) |
\(\approx\) |
\(0.8670174799 - 0.8154241131i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1259 | \( 1 \) |
good | 2 | \( 1 + (0.405 - 0.914i)T \) |
| 3 | \( 1 + (-0.0174 - 0.999i)T \) |
| 5 | \( 1 + (0.961 + 0.276i)T \) |
| 7 | \( 1 + (0.472 - 0.881i)T \) |
| 11 | \( 1 + (-0.598 - 0.801i)T \) |
| 13 | \( 1 + (-0.418 + 0.908i)T \) |
| 17 | \( 1 + (-0.902 + 0.430i)T \) |
| 19 | \( 1 + (-0.498 + 0.866i)T \) |
| 23 | \( 1 + (0.690 - 0.723i)T \) |
| 29 | \( 1 + (0.594 + 0.803i)T \) |
| 31 | \( 1 + (-0.532 + 0.846i)T \) |
| 37 | \( 1 + (-0.0274 + 0.999i)T \) |
| 41 | \( 1 + (0.967 + 0.251i)T \) |
| 43 | \( 1 + (-0.117 - 0.993i)T \) |
| 47 | \( 1 + (-0.732 - 0.681i)T \) |
| 53 | \( 1 + (0.718 + 0.695i)T \) |
| 59 | \( 1 + (-0.802 - 0.596i)T \) |
| 61 | \( 1 + (0.765 - 0.643i)T \) |
| 67 | \( 1 + (-0.377 + 0.926i)T \) |
| 71 | \( 1 + (0.990 + 0.139i)T \) |
| 73 | \( 1 + (-0.735 + 0.677i)T \) |
| 79 | \( 1 + (0.181 - 0.983i)T \) |
| 83 | \( 1 + (-0.602 + 0.798i)T \) |
| 89 | \( 1 + (0.459 + 0.888i)T \) |
| 97 | \( 1 + (-0.999 + 0.0149i)T \) |
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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
−21.296251279084967315828668382195, −20.57981246102287588002336301670, −19.56074432859222408180731983505, −18.020936318605271428216902536357, −17.766312895612637318687563703014, −17.1796907721415933554703424170, −16.15916823986761073630215495624, −15.39740283030171246001087932552, −15.05137205446664850816804302095, −14.27089804220475480660215226367, −13.22921130610750619932722783748, −12.759706900835940239653496354591, −11.64103174657483140093007542448, −10.698726187557087257723130039115, −9.573765725915055984992955881879, −9.23052706185164561378156295081, −8.36785424127197462672182172137, −7.453455076308392279537444498395, −6.24693901888753635085472180462, −5.51473264121043435157153597601, −4.93518282362665865609323015478, −4.370367441615048201801386574205, −2.85026012166489222806471316121, −2.33064264522026723022612799782, −0.28041529411405397323395964754,
0.95937687792399412518271654387, 1.74829344350500667980197107354, 2.45777167626717507502131689261, 3.41841962314089852939177747010, 4.62333118414420863876061085011, 5.44295450106135337062941575815, 6.41321506899119234094175827525, 6.97745673015506897289179785536, 8.35154615345368445773250269681, 8.90494155214873469335222807682, 10.1471954838075705981849024628, 10.78143505037736396105388236162, 11.33269963259253000028131801465, 12.44606629256127697141595709275, 13.0542640385168818497368557380, 13.78259716294574236321073905131, 14.210857153791695900993312938444, 14.86705253432140207472746830878, 16.53492921804915936516713042023, 17.21615717015172008512862844241, 17.97537860373994151436715689202, 18.60674196849631047878077514498, 19.22418482153687816690089878093, 20.073018774757010410110455048478, 20.79057124885581431065524800746