L(s) = 1 | + (−0.981 − 0.189i)2-s + (0.928 + 0.371i)4-s + (0.0475 + 0.998i)5-s + (−0.235 − 0.971i)7-s + (−0.841 − 0.540i)8-s + (0.142 − 0.989i)10-s + (−0.928 − 0.371i)13-s + (0.0475 + 0.998i)14-s + (0.723 + 0.690i)16-s + (−0.415 + 0.909i)17-s + (−0.415 − 0.909i)19-s + (−0.327 + 0.945i)20-s + (0.235 − 0.971i)23-s + (−0.995 + 0.0950i)25-s + (0.841 + 0.540i)26-s + ⋯ |
L(s) = 1 | + (−0.981 − 0.189i)2-s + (0.928 + 0.371i)4-s + (0.0475 + 0.998i)5-s + (−0.235 − 0.971i)7-s + (−0.841 − 0.540i)8-s + (0.142 − 0.989i)10-s + (−0.928 − 0.371i)13-s + (0.0475 + 0.998i)14-s + (0.723 + 0.690i)16-s + (−0.415 + 0.909i)17-s + (−0.415 − 0.909i)19-s + (−0.327 + 0.945i)20-s + (0.235 − 0.971i)23-s + (−0.995 + 0.0950i)25-s + (0.841 + 0.540i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s+1) \, L(s)\cr =\mathstrut & (0.999 + 0.0187i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.7339594470 + 0.006881611396i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.7339594470 + 0.006881611396i\) |
\(L(1)\) |
\(\approx\) |
\(0.5864136146 + 0.005807667497i\) |
\(L(1)\) |
\(\approx\) |
\(0.5864136146 + 0.005807667497i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.981 - 0.189i)T \) |
| 5 | \( 1 + (0.0475 + 0.998i)T \) |
| 7 | \( 1 + (-0.235 - 0.971i)T \) |
| 13 | \( 1 + (-0.928 - 0.371i)T \) |
| 17 | \( 1 + (-0.415 + 0.909i)T \) |
| 19 | \( 1 + (-0.415 - 0.909i)T \) |
| 23 | \( 1 + (0.235 - 0.971i)T \) |
| 29 | \( 1 + (-0.580 + 0.814i)T \) |
| 31 | \( 1 + (-0.786 - 0.618i)T \) |
| 37 | \( 1 + (-0.142 + 0.989i)T \) |
| 41 | \( 1 + (0.327 + 0.945i)T \) |
| 43 | \( 1 + (-0.0475 + 0.998i)T \) |
| 47 | \( 1 + (-0.327 + 0.945i)T \) |
| 53 | \( 1 + (-0.959 - 0.281i)T \) |
| 59 | \( 1 + (0.981 - 0.189i)T \) |
| 61 | \( 1 + (0.327 - 0.945i)T \) |
| 67 | \( 1 + (-0.327 - 0.945i)T \) |
| 71 | \( 1 + (0.415 + 0.909i)T \) |
| 73 | \( 1 + (0.959 - 0.281i)T \) |
| 79 | \( 1 + (0.888 + 0.458i)T \) |
| 83 | \( 1 + (-0.235 - 0.971i)T \) |
| 89 | \( 1 + (0.415 - 0.909i)T \) |
| 97 | \( 1 + (0.0475 - 0.998i)T \) |
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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
−21.06652865269110278401793115800, −20.385966179829678395184134370142, −19.4668055500983474635786086879, −19.03543000385095610656645974366, −18.05446940355950899805252292892, −17.37996498579648350749981496408, −16.58114407712340607044408450004, −15.99140731681720689514047259374, −15.261137098650944350577749846500, −14.42033319716946343196371994123, −13.26753590347991739096702485978, −12.1609331903376216132568586309, −11.94294933097888593014228242238, −10.81689170463672062889055579759, −9.62204072221680622139138568442, −9.30461199400435563688580308302, −8.54452121602484556867160337643, −7.65203571322047100529785472992, −6.81121750094946390150636440944, −5.61502950237471579306046450201, −5.23246961027032487183729597446, −3.79075922971667375717921196736, −2.38236697650381924689619508018, −1.79976606033126954124171367460, −0.441142840752585206362267499566,
0.43521085252073846709070986017, 1.77176793340653210375200325549, 2.74039446494301828729298803004, 3.50845224049405244167071001622, 4.65766879567294840541421573306, 6.23246172400323487684083334590, 6.773370314499454879648261687361, 7.52350084133861709353833026367, 8.2729609589017513784368759611, 9.47850476414537035342624113788, 10.05365713261958695242133759201, 10.96915845166684086639997228932, 11.15533945308184010394839261547, 12.584261110756100218038814667, 13.1517096354312482485726824707, 14.51399948071533740751144926559, 14.936588371076150189226486831949, 15.91578600588451893430227930003, 16.88939636290500605195577338884, 17.35613545605161270929322353716, 18.13383458362872143586653321631, 18.944637233783078118774278098219, 19.67798232419277928622587291671, 20.11207652733798749641487846378, 21.1309436483552111147598226480