L(s) = 1 | + (0.710 − 0.703i)2-s + (0.104 + 0.994i)3-s + (0.00951 − 0.999i)4-s + (0.774 + 0.633i)6-s + (−0.696 − 0.717i)8-s + (−0.978 + 0.207i)9-s + (0.995 − 0.0950i)12-s + (0.736 + 0.676i)13-s + (−0.999 − 0.0190i)16-s + (−0.879 − 0.475i)17-s + (−0.548 + 0.836i)18-s + (−0.991 − 0.132i)19-s + (0.327 − 0.945i)23-s + (0.640 − 0.768i)24-s + (0.999 − 0.0380i)26-s + (−0.309 − 0.951i)27-s + ⋯ |
L(s) = 1 | + (0.710 − 0.703i)2-s + (0.104 + 0.994i)3-s + (0.00951 − 0.999i)4-s + (0.774 + 0.633i)6-s + (−0.696 − 0.717i)8-s + (−0.978 + 0.207i)9-s + (0.995 − 0.0950i)12-s + (0.736 + 0.676i)13-s + (−0.999 − 0.0190i)16-s + (−0.879 − 0.475i)17-s + (−0.548 + 0.836i)18-s + (−0.991 − 0.132i)19-s + (0.327 − 0.945i)23-s + (0.640 − 0.768i)24-s + (0.999 − 0.0380i)26-s + (−0.309 − 0.951i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4235 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.882 - 0.470i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.253071113 - 0.5628986956i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.253071113 - 0.5628986956i\) |
\(L(1)\) |
\(\approx\) |
\(1.442705344 - 0.2360354976i\) |
\(L(1)\) |
\(\approx\) |
\(1.442705344 - 0.2360354976i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 \) |
| 7 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (0.710 - 0.703i)T \) |
| 3 | \( 1 + (0.104 + 0.994i)T \) |
| 13 | \( 1 + (0.736 + 0.676i)T \) |
| 17 | \( 1 + (-0.879 - 0.475i)T \) |
| 19 | \( 1 + (-0.991 - 0.132i)T \) |
| 23 | \( 1 + (0.327 - 0.945i)T \) |
| 29 | \( 1 + (0.941 + 0.336i)T \) |
| 31 | \( 1 + (-0.398 + 0.917i)T \) |
| 37 | \( 1 + (0.948 - 0.318i)T \) |
| 41 | \( 1 + (-0.362 + 0.931i)T \) |
| 43 | \( 1 + (0.142 + 0.989i)T \) |
| 47 | \( 1 + (-0.548 - 0.836i)T \) |
| 53 | \( 1 + (0.999 - 0.0190i)T \) |
| 59 | \( 1 + (0.988 + 0.151i)T \) |
| 61 | \( 1 + (0.964 - 0.263i)T \) |
| 67 | \( 1 + (-0.0475 - 0.998i)T \) |
| 71 | \( 1 + (-0.564 - 0.825i)T \) |
| 73 | \( 1 + (0.905 + 0.424i)T \) |
| 79 | \( 1 + (-0.0665 - 0.997i)T \) |
| 83 | \( 1 + (0.921 + 0.389i)T \) |
| 89 | \( 1 + (0.235 + 0.971i)T \) |
| 97 | \( 1 + (0.985 + 0.170i)T \) |
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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
−18.269408264129706803777668837313, −17.51186180691867839818552560021, −17.28755613635402505543860015443, −16.371530490396237694125508756133, −15.50081046612778461987243234546, −15.06506312851363636667670611365, −14.30213819728361902244175694725, −13.538832448680994548667729866039, −13.065522256980941328305984261706, −12.68377527336878457463020318674, −11.68283005581599044477934604808, −11.262338586800345713773516727171, −10.29723800647993062242980115681, −9.01431354070677795696645920177, −8.50825016450518137608116310998, −7.91565968925481055761890263960, −7.16829690206689603171209399245, −6.480125090909409908136438222, −5.926922709118223237391361860298, −5.285266944400382079678975177465, −4.18979602086800401806161301713, −3.57128406947370397599096599877, −2.60155192192132884887905067628, −1.984729349542275687868197665957, −0.733278675496003672746240645055,
0.67393712799392678754896493390, 1.92275605207944574651631781359, 2.63284279851684761897909679364, 3.39122127388839400092366748311, 4.21798792219628650363705884238, 4.64478921805646385997581274500, 5.34023749245911530409127183300, 6.42700702739029809223391379121, 6.6712200293392193918968871630, 8.25056186692229216112947646874, 8.88449629349869822429241338731, 9.42420825096735601309262798289, 10.33118161427997712989246316218, 10.80739081878055312225145219122, 11.384709665065752289855854951451, 12.03970599154515384394136924622, 13.015099328044407998444900241905, 13.50622516876933890220745999261, 14.394095813965290334691765972894, 14.74531846485838094331166140978, 15.50723861970759810928030634688, 16.21855744450975895725244131873, 16.637451296064538106315190429405, 17.855984994798470691406571288403, 18.30864238880431964889782115316