L(s) = 1 | + (−0.969 + 0.244i)2-s + (0.879 − 0.475i)4-s + (0.861 + 0.508i)5-s + (0.161 + 0.986i)7-s + (−0.736 + 0.676i)8-s + (−0.959 − 0.281i)10-s + (−0.991 − 0.132i)13-s + (−0.398 − 0.917i)14-s + (0.548 − 0.836i)16-s + (0.0855 + 0.996i)17-s + (−0.921 − 0.389i)19-s + (0.999 + 0.0380i)20-s + (−0.888 − 0.458i)23-s + (0.483 + 0.875i)25-s + (0.993 − 0.113i)26-s + ⋯ |
L(s) = 1 | + (−0.969 + 0.244i)2-s + (0.879 − 0.475i)4-s + (0.861 + 0.508i)5-s + (0.161 + 0.986i)7-s + (−0.736 + 0.676i)8-s + (−0.959 − 0.281i)10-s + (−0.991 − 0.132i)13-s + (−0.398 − 0.917i)14-s + (0.548 − 0.836i)16-s + (0.0855 + 0.996i)17-s + (−0.921 − 0.389i)19-s + (0.999 + 0.0380i)20-s + (−0.888 − 0.458i)23-s + (0.483 + 0.875i)25-s + (0.993 − 0.113i)26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1089 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.941 - 0.336i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(-0.04710677354 + 0.2717375265i\) |
\(L(\frac12)\) |
\(\approx\) |
\(-0.04710677354 + 0.2717375265i\) |
\(L(1)\) |
\(\approx\) |
\(0.5764307875 + 0.2155181728i\) |
\(L(1)\) |
\(\approx\) |
\(0.5764307875 + 0.2155181728i\) |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-0.969 + 0.244i)T \) |
| 5 | \( 1 + (0.861 + 0.508i)T \) |
| 7 | \( 1 + (0.161 + 0.986i)T \) |
| 13 | \( 1 + (-0.991 - 0.132i)T \) |
| 17 | \( 1 + (0.0855 + 0.996i)T \) |
| 19 | \( 1 + (-0.921 - 0.389i)T \) |
| 23 | \( 1 + (-0.888 - 0.458i)T \) |
| 29 | \( 1 + (-0.999 + 0.0190i)T \) |
| 31 | \( 1 + (-0.761 - 0.647i)T \) |
| 37 | \( 1 + (-0.564 + 0.825i)T \) |
| 41 | \( 1 + (-0.830 - 0.556i)T \) |
| 43 | \( 1 + (-0.995 - 0.0950i)T \) |
| 47 | \( 1 + (0.345 - 0.938i)T \) |
| 53 | \( 1 + (-0.998 + 0.0570i)T \) |
| 59 | \( 1 + (-0.0665 - 0.997i)T \) |
| 61 | \( 1 + (0.272 + 0.962i)T \) |
| 67 | \( 1 + (-0.786 + 0.618i)T \) |
| 71 | \( 1 + (0.974 - 0.226i)T \) |
| 73 | \( 1 + (-0.254 - 0.967i)T \) |
| 79 | \( 1 + (-0.948 - 0.318i)T \) |
| 83 | \( 1 + (0.988 + 0.151i)T \) |
| 89 | \( 1 + (-0.654 - 0.755i)T \) |
| 97 | \( 1 + (0.861 - 0.508i)T \) |
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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
−20.806022426201933115735353545328, −20.17180245226146421337011840532, −19.599977811621349841914095990385, −18.55734327028470736349289777543, −17.82892550490616563876329442864, −17.10442027607758824714020162151, −16.68618764565912223998738996112, −15.907246425000311337236537943977, −14.67857972512533757653941700111, −13.95148437665537494125302658306, −12.97058754589083968930626210710, −12.27174800860677347595288962430, −11.29747536747903514386620039051, −10.43009870023200447617312613771, −9.784124891546303696662555827487, −9.18706836944250311634813637556, −8.15594355763101867803454308558, −7.35949124209190179177946819492, −6.615696336355653560217281498163, −5.52628153933735263506345881414, −4.4884746613050198420408282414, −3.35663419571070250359287689029, −2.14034007398577327228063050970, −1.48379522140074128018032302219, −0.13937326671123440771999347594,
1.94300516936216922161386773553, 2.08427481546632758970300015471, 3.30974632885879586986867851306, 4.98096413785451732630044776843, 5.834527441004612875065896452982, 6.44049155509142663236014090809, 7.37219119768913328812226277333, 8.37284933883733519570013348082, 9.03084159094433665630381119529, 9.92843773800048693166550825115, 10.452984356259919562797961450732, 11.40602040978154906904710906079, 12.25969914018013176304815419521, 13.1518438978424543769607586954, 14.45012544330156376758816998991, 14.917671688860458023764773904684, 15.51498039111913725690193702046, 16.820862305771046751617442476901, 17.16723472999758714167002714115, 18.04348629971395204595302415936, 18.67298233499262978321523171809, 19.24651492430531037072548894142, 20.21340354662798126976700585889, 21.05087803270213590514981138155, 21.91446501664155613789387443151