L(s) = 1 | + (0.767 − 0.640i)2-s + (0.901 + 0.432i)3-s + (0.179 − 0.983i)4-s + (−0.726 + 0.687i)5-s + (0.969 − 0.245i)6-s + (0.948 − 0.317i)7-s + (−0.492 − 0.870i)8-s + (0.626 + 0.779i)9-s + (−0.117 + 0.993i)10-s + (−0.691 + 0.722i)11-s + (0.586 − 0.809i)12-s + (0.827 + 0.561i)13-s + (0.524 − 0.851i)14-s + (−0.952 + 0.305i)15-s + (−0.935 − 0.352i)16-s + (0.999 + 0.0248i)17-s + ⋯ |
L(s) = 1 | + (0.767 − 0.640i)2-s + (0.901 + 0.432i)3-s + (0.179 − 0.983i)4-s + (−0.726 + 0.687i)5-s + (0.969 − 0.245i)6-s + (0.948 − 0.317i)7-s + (−0.492 − 0.870i)8-s + (0.626 + 0.779i)9-s + (−0.117 + 0.993i)10-s + (−0.691 + 0.722i)11-s + (0.586 − 0.809i)12-s + (0.827 + 0.561i)13-s + (0.524 − 0.851i)14-s + (−0.952 + 0.305i)15-s + (−0.935 − 0.352i)16-s + (0.999 + 0.0248i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.984 - 0.176i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(2.775562736 - 0.2463958463i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.775562736 - 0.2463958463i\) |
\(L(1)\) |
\(\approx\) |
\(2.009313659 - 0.2551624075i\) |
\(L(1)\) |
\(\approx\) |
\(2.009313659 - 0.2551624075i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (0.767 - 0.640i)T \) |
| 3 | \( 1 + (0.901 + 0.432i)T \) |
| 5 | \( 1 + (-0.726 + 0.687i)T \) |
| 7 | \( 1 + (0.948 - 0.317i)T \) |
| 11 | \( 1 + (-0.691 + 0.722i)T \) |
| 13 | \( 1 + (0.827 + 0.561i)T \) |
| 17 | \( 1 + (0.999 + 0.0248i)T \) |
| 19 | \( 1 + (-0.556 + 0.831i)T \) |
| 29 | \( 1 + (0.867 - 0.498i)T \) |
| 31 | \( 1 + (0.751 - 0.659i)T \) |
| 37 | \( 1 + (-0.977 - 0.209i)T \) |
| 41 | \( 1 + (-0.404 - 0.914i)T \) |
| 43 | \( 1 + (0.0310 + 0.999i)T \) |
| 47 | \( 1 + (-0.990 + 0.136i)T \) |
| 53 | \( 1 + (-0.117 - 0.993i)T \) |
| 59 | \( 1 + (-0.0186 - 0.999i)T \) |
| 61 | \( 1 + (0.995 + 0.0991i)T \) |
| 67 | \( 1 + (-0.191 + 0.981i)T \) |
| 71 | \( 1 + (-0.999 + 0.0124i)T \) |
| 73 | \( 1 + (0.867 + 0.498i)T \) |
| 79 | \( 1 + (0.251 - 0.967i)T \) |
| 83 | \( 1 + (0.890 - 0.454i)T \) |
| 89 | \( 1 + (-0.873 + 0.487i)T \) |
| 97 | \( 1 + (0.130 - 0.991i)T \) |
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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
−23.71173376246444233319316204535, −23.11739536513870705492010666040, −21.5548436436048499276745806939, −21.04343653819707728968689321280, −20.4272441190662477600722703341, −19.40738045174385653935786574242, −18.402422973903238703305698478009, −17.59197344994461424905026167202, −16.42318780017517560998028486162, −15.5564959713228558022003581334, −15.10071095735108080395469506844, −14.020239724387919826325797256498, −13.40092815284919100371247141851, −12.49436498593493121676329158073, −11.819053883346753226343688174058, −10.700402648968873240558749949352, −8.8713883395066994229856232130, −8.29189135254305775828208224698, −7.89529042439158332777578504380, −6.75670403835536657932841118098, −5.48756733366183692405040054049, −4.68443953468048398476122708937, −3.55589447681003658229439217071, −2.77145973220113671619006716260, −1.24149262715727752375322020465,
1.55910325437914872139352806237, 2.517668454218434198416119845594, 3.61621527528357156012819392657, 4.22783532466432702158011634092, 5.14441637069097221955325531631, 6.60489904914114381496205136178, 7.721404581063367965675733314401, 8.41872067759731131774387471565, 9.93059238949878089365791025918, 10.44583794356343495933852802757, 11.33287380355271621467238215631, 12.209218464746394518804633400443, 13.33276015071337391512779603295, 14.24933515715805192102551141580, 14.66109059194349972793203621170, 15.48438711365293443217424742959, 16.22542150417350618584843649260, 17.86568802459240922862115829822, 18.910021103202654352422347710266, 19.226130283199953592306355141502, 20.49434057715068284677813284266, 20.85700891567829218967532018973, 21.47986565645439896996956264810, 22.71025573406954473217772374600, 23.3125991452995060971767388287