L(s) = 1 | + (0.5 − 0.866i)2-s + (−0.499 − 0.866i)4-s − 0.999·8-s + 3·11-s + (1 − 1.73i)13-s + (−0.5 + 0.866i)16-s + (1.5 − 2.59i)17-s + (−0.5 − 0.866i)19-s + (1.5 − 2.59i)22-s + 6·23-s − 5·25-s + (−0.999 − 1.73i)26-s + (3 + 5.19i)29-s + (−2 − 3.46i)31-s + (0.499 + 0.866i)32-s + ⋯ |
L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s − 0.353·8-s + 0.904·11-s + (0.277 − 0.480i)13-s + (−0.125 + 0.216i)16-s + (0.363 − 0.630i)17-s + (−0.114 − 0.198i)19-s + (0.319 − 0.553i)22-s + 1.25·23-s − 25-s + (−0.196 − 0.339i)26-s + (0.557 + 0.964i)29-s + (−0.359 − 0.622i)31-s + (0.0883 + 0.153i)32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2646 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.0788 + 0.996i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.204833154\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.204833154\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 + 0.866i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + 5T^{2} \) |
| 11 | \( 1 - 3T + 11T^{2} \) |
| 13 | \( 1 + (-1 + 1.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 + (-1.5 + 2.59i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.5 + 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 6T + 23T^{2} \) |
| 29 | \( 1 + (-3 - 5.19i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 + 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-2 - 3.46i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.5 - 7.79i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-0.5 - 0.866i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-3 + 5.19i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-6 + 10.3i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 + 2.59i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4 + 6.92i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (2.5 + 4.33i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12T + 71T^{2} \) |
| 73 | \( 1 + (-5.5 + 9.52i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 + 3.46i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (6 + 10.3i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (3 + 5.19i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-2.5 - 4.33i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−8.801068843666960516803729481475, −8.005442320517725165305069637363, −6.99644987445691438954436417764, −6.33528618852398443715687928230, −5.36060665644419951709524121185, −4.70960741806288521832459952492, −3.66462531008247542517508268212, −3.04244942793068378935527259836, −1.84113257286933209092450952087, −0.74444858724157812810508037784,
1.18747223031186645529145946301, 2.50004754535446845544625641126, 3.77548366728852827525990345450, 4.16514106222346808771996634468, 5.31950809387760137887153122025, 5.99363409826546343911512309911, 6.75496875614406065778576872497, 7.38694678498765606700225856026, 8.297868267920520617097101934547, 8.934572609032773817426846039577