L(s) = 1 | + (−0.977 − 0.209i)2-s + (−0.935 + 0.352i)3-s + (0.912 + 0.409i)4-s + (0.984 + 0.172i)5-s + (0.988 − 0.148i)6-s + (−0.907 − 0.421i)7-s + (−0.806 − 0.591i)8-s + (0.751 − 0.659i)9-s + (−0.926 − 0.375i)10-s + (−0.885 + 0.465i)11-s + (−0.998 − 0.0620i)12-s + (−0.0434 + 0.999i)13-s + (0.798 + 0.601i)14-s + (−0.982 + 0.185i)15-s + (0.664 + 0.747i)16-s + (0.323 − 0.946i)17-s + ⋯ |
L(s) = 1 | + (−0.977 − 0.209i)2-s + (−0.935 + 0.352i)3-s + (0.912 + 0.409i)4-s + (0.984 + 0.172i)5-s + (0.988 − 0.148i)6-s + (−0.907 − 0.421i)7-s + (−0.806 − 0.591i)8-s + (0.751 − 0.659i)9-s + (−0.926 − 0.375i)10-s + (−0.885 + 0.465i)11-s + (−0.998 − 0.0620i)12-s + (−0.0434 + 0.999i)13-s + (0.798 + 0.601i)14-s + (−0.982 + 0.185i)15-s + (0.664 + 0.747i)16-s + (0.323 − 0.946i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 529 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0118i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.6219372499 + 0.003693566098i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.6219372499 + 0.003693566098i\) |
\(L(1)\) |
\(\approx\) |
\(0.5748241343 + 0.007620089448i\) |
\(L(1)\) |
\(\approx\) |
\(0.5748241343 + 0.007620089448i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 23 | \( 1 \) |
good | 2 | \( 1 + (-0.977 - 0.209i)T \) |
| 3 | \( 1 + (-0.935 + 0.352i)T \) |
| 5 | \( 1 + (0.984 + 0.172i)T \) |
| 7 | \( 1 + (-0.907 - 0.421i)T \) |
| 11 | \( 1 + (-0.885 + 0.465i)T \) |
| 13 | \( 1 + (-0.0434 + 0.999i)T \) |
| 17 | \( 1 + (0.323 - 0.946i)T \) |
| 19 | \( 1 + (0.346 - 0.938i)T \) |
| 29 | \( 1 + (0.586 + 0.809i)T \) |
| 31 | \( 1 + (-0.117 - 0.993i)T \) |
| 37 | \( 1 + (-0.426 + 0.904i)T \) |
| 41 | \( 1 + (0.369 - 0.929i)T \) |
| 43 | \( 1 + (-0.0186 - 0.999i)T \) |
| 47 | \( 1 + (0.854 + 0.519i)T \) |
| 53 | \( 1 + (-0.926 + 0.375i)T \) |
| 59 | \( 1 + (-0.596 - 0.802i)T \) |
| 61 | \( 1 + (0.251 + 0.967i)T \) |
| 67 | \( 1 + (0.980 - 0.197i)T \) |
| 71 | \( 1 + (0.813 + 0.581i)T \) |
| 73 | \( 1 + (0.586 - 0.809i)T \) |
| 79 | \( 1 + (-0.986 + 0.160i)T \) |
| 83 | \( 1 + (0.0310 - 0.999i)T \) |
| 89 | \( 1 + (0.948 + 0.317i)T \) |
| 97 | \( 1 + (-0.972 + 0.233i)T \) |
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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
−23.51122287758133952910035084798, −22.809197406096030121290485594884, −21.646621450524440831025076105896, −21.118492273737366125822802416938, −19.869686289558818714398976984866, −18.96105001549219959455057891301, −18.27592636945252656217888244088, −17.65556514785026732407973127951, −16.822729444233816690533795651420, −16.1346436813619127717683672041, −15.41204226960055050275528210002, −14.04273750871654838985884970403, −12.7911522704108623641256621459, −12.45855213580399135949148500140, −11.06568085925479284796476503316, −10.21829140380121162482313239682, −9.85121924417858954830349813252, −8.51675127829773517570403055279, −7.67111184326021427828561444759, −6.42332506371815471419175271940, −5.87593526625734318023024404561, −5.265934307965065127085015339968, −3.11057608759956606997461830114, −2.0108851903341716935543722912, −0.81351807288670364807314449714,
0.75416144438266889804355340755, 2.16343393956664805743395855742, 3.24043670040564262838965284421, 4.74002327170250477790304818984, 5.84412030727418437113189284833, 6.816686898223618775440713217527, 7.27746941267119536294332515894, 9.09235783540597946754220063150, 9.63700116024939651536011979145, 10.32592125735650081143714321269, 11.06329921039911053980102255712, 12.10503476292747933491639322728, 12.948970083429159980989593187296, 13.96267560699829156820382902969, 15.49295090996141022236769955373, 16.06093568191537029157758513741, 16.93255673896961060225669187875, 17.462974370325063215430884121919, 18.41992437048088478232291509835, 18.88643298783567963202335751277, 20.29408280365896015633490239139, 20.86357035615700922769036326561, 21.8118686603821723440294743782, 22.38686953490774616813651378189, 23.53087859286411386052567459317