| L(s) = 1 | + (−0.978 − 0.207i)2-s + (0.199 + 0.615i)3-s + (0.913 + 0.406i)4-s + (−0.382 − 3.63i)5-s + (−0.0676 − 0.643i)6-s + (1.22 + 1.36i)7-s + (−0.809 − 0.587i)8-s + (2.08 − 1.51i)9-s + (−0.382 + 3.63i)10-s + 3.01·11-s + (−0.0676 + 0.643i)12-s + (−0.306 + 0.530i)13-s + (−0.915 − 1.58i)14-s + (2.16 − 0.962i)15-s + (0.669 + 0.743i)16-s + (−2.83 − 1.26i)17-s + ⋯ |
| L(s) = 1 | + (−0.691 − 0.147i)2-s + (0.115 + 0.355i)3-s + (0.456 + 0.203i)4-s + (−0.170 − 1.62i)5-s + (−0.0276 − 0.262i)6-s + (0.463 + 0.514i)7-s + (−0.286 − 0.207i)8-s + (0.696 − 0.505i)9-s + (−0.120 + 1.15i)10-s + 0.908·11-s + (−0.0195 + 0.185i)12-s + (−0.0849 + 0.147i)13-s + (−0.244 − 0.423i)14-s + (0.558 − 0.248i)15-s + (0.167 + 0.185i)16-s + (−0.686 − 0.305i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 122 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.817 + 0.575i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.823517 - 0.260929i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.823517 - 0.260929i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.978 + 0.207i)T \) |
| 61 | \( 1 + (-6.40 + 4.47i)T \) |
| good | 3 | \( 1 + (-0.199 - 0.615i)T + (-2.42 + 1.76i)T^{2} \) |
| 5 | \( 1 + (0.382 + 3.63i)T + (-4.89 + 1.03i)T^{2} \) |
| 7 | \( 1 + (-1.22 - 1.36i)T + (-0.731 + 6.96i)T^{2} \) |
| 11 | \( 1 - 3.01T + 11T^{2} \) |
| 13 | \( 1 + (0.306 - 0.530i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (2.83 + 1.26i)T + (11.3 + 12.6i)T^{2} \) |
| 19 | \( 1 + (-0.666 + 0.740i)T + (-1.98 - 18.8i)T^{2} \) |
| 23 | \( 1 + (2.91 - 2.12i)T + (7.10 - 21.8i)T^{2} \) |
| 29 | \( 1 + (-2.21 - 3.83i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (9.36 - 1.99i)T + (28.3 - 12.6i)T^{2} \) |
| 37 | \( 1 + (3.51 - 10.8i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (0.770 - 2.37i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 + (-9.47 + 4.21i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (0.641 + 1.11i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (6.73 + 4.89i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.56 - 0.970i)T + (53.8 + 23.9i)T^{2} \) |
| 67 | \( 1 + (-0.979 - 9.31i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (1.60 - 15.3i)T + (-69.4 - 14.7i)T^{2} \) |
| 73 | \( 1 + (1.17 - 11.1i)T + (-71.4 - 15.1i)T^{2} \) |
| 79 | \( 1 + (-11.1 + 4.95i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (3.27 + 0.695i)T + (75.8 + 33.7i)T^{2} \) |
| 89 | \( 1 + (5.52 + 17.0i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (2.05 - 0.436i)T + (88.6 - 39.4i)T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.03157181062755247146530208746, −12.18503571345910130464592137804, −11.42891011474938947574979474529, −9.840052037172089624213855932758, −9.032478093940500841906870317474, −8.454764837385547116631357058037, −6.93986246163725696996615674368, −5.21710358000979867245179740083, −4.00430145380075354433759321767, −1.46922092997132159086380067454,
2.12099490707370733916489732345, 3.98159471724408456416909146720, 6.23951113078191991240912145198, 7.20040680952653901104620430834, 7.82884645524217169749019594911, 9.406210952692659098984946902392, 10.67263721640611492505546783194, 10.99770871448929790290146991162, 12.37678682208878824840379221104, 13.89157921357166866235676188140
