L(s) = 1 | + (−0.673 + 0.739i)2-s + (−0.445 − 0.895i)3-s + (−0.0922 − 0.995i)4-s + (−0.273 + 0.961i)5-s + (0.961 + 0.273i)6-s + (0.361 − 0.932i)7-s + (0.798 + 0.602i)8-s + (−0.602 + 0.798i)9-s + (−0.526 − 0.850i)10-s + (−0.850 − 0.526i)11-s + (−0.850 + 0.526i)12-s + (0.673 + 0.739i)13-s + (0.445 + 0.895i)14-s + (0.982 − 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.273 − 0.961i)17-s + ⋯ |
L(s) = 1 | + (−0.673 + 0.739i)2-s + (−0.445 − 0.895i)3-s + (−0.0922 − 0.995i)4-s + (−0.273 + 0.961i)5-s + (0.961 + 0.273i)6-s + (0.361 − 0.932i)7-s + (0.798 + 0.602i)8-s + (−0.602 + 0.798i)9-s + (−0.526 − 0.850i)10-s + (−0.850 − 0.526i)11-s + (−0.850 + 0.526i)12-s + (0.673 + 0.739i)13-s + (0.445 + 0.895i)14-s + (0.982 − 0.183i)15-s + (−0.982 + 0.183i)16-s + (0.273 − 0.961i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1021 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (-0.913 + 0.405i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.01091988400 + 0.05149502434i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.01091988400 + 0.05149502434i\) |
\(L(1)\) |
\(\approx\) |
\(0.5592763483 + 0.006513429636i\) |
\(L(1)\) |
\(\approx\) |
\(0.5592763483 + 0.006513429636i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 1021 | \( 1 \) |
good | 2 | \( 1 + (-0.673 + 0.739i)T \) |
| 3 | \( 1 + (-0.445 - 0.895i)T \) |
| 5 | \( 1 + (-0.273 + 0.961i)T \) |
| 7 | \( 1 + (0.361 - 0.932i)T \) |
| 11 | \( 1 + (-0.850 - 0.526i)T \) |
| 13 | \( 1 + (0.673 + 0.739i)T \) |
| 17 | \( 1 + (0.273 - 0.961i)T \) |
| 19 | \( 1 + (0.798 - 0.602i)T \) |
| 23 | \( 1 + (-0.982 - 0.183i)T \) |
| 29 | \( 1 + (-0.932 + 0.361i)T \) |
| 31 | \( 1 + (0.798 + 0.602i)T \) |
| 37 | \( 1 + (-0.895 + 0.445i)T \) |
| 41 | \( 1 + (0.602 + 0.798i)T \) |
| 43 | \( 1 + (0.361 - 0.932i)T \) |
| 47 | \( 1 + (0.445 + 0.895i)T \) |
| 53 | \( 1 + (0.183 - 0.982i)T \) |
| 59 | \( 1 + (-0.895 + 0.445i)T \) |
| 61 | \( 1 + (-0.445 - 0.895i)T \) |
| 67 | \( 1 + T \) |
| 71 | \( 1 + (-0.602 + 0.798i)T \) |
| 73 | \( 1 + (0.445 - 0.895i)T \) |
| 79 | \( 1 + (0.850 - 0.526i)T \) |
| 83 | \( 1 + (-0.445 - 0.895i)T \) |
| 89 | \( 1 + (0.0922 - 0.995i)T \) |
| 97 | \( 1 + (-0.361 + 0.932i)T \) |
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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
−20.88812711822212673695697726371, −20.56291011812642150631842717377, −19.70083271382648799153695613873, −18.61555557021414939639318224000, −17.91701655301662149547840509989, −17.27338260059631157830994516095, −16.417359415720436139321276055748, −15.65559025601908039206928269838, −15.25987543580571441820023755485, −13.770454142185945937654310154314, −12.593671657206493650974260331800, −12.267280752878689518779067980979, −11.41442211507814251921169453625, −10.584736930280639188522076493244, −9.8185400843239320422214409632, −9.101328595107570845062317518239, −8.22463672373538997388349757328, −7.77148674635169935620126019691, −5.88158704721676580081081114818, −5.337315917732213970641270709605, −4.2512706775211678777014867087, −3.54794409593843167466056648156, −2.32918207716851723003857129278, −1.21438010230514768231140981106, −0.01968839055920416616114426222,
0.87517095684588251578888981224, 1.962583741120110135511499163728, 3.16550193157879575412440789795, 4.61180353339551272531086281886, 5.59068192345310311443392364847, 6.48006296834704960751436870083, 7.172463855395289405247135409246, 7.6869104455118564544654547804, 8.44800469822505405663322486791, 9.71596116299148683639582735587, 10.72423970238601278828007504956, 11.10078289770277983769021102564, 11.92571216938867178128729020104, 13.55687131931524978377632922375, 13.80308534905546742849536720210, 14.494618073491378473531750926639, 15.8533617022867489249218071846, 16.18554156613051230042020680330, 17.22496412677803225741108179107, 17.970945272096135372352310901900, 18.47277409637670817648542624241, 19.02520258720999723051342803198, 19.89887410193639396468958104605, 20.725652539566465590058518967241, 22.07445375911377435241293689548