L(s) = 1 | + (0.5 + 0.866i)2-s + (0.5 − 0.866i)3-s + (−0.499 + 0.866i)4-s + 0.999·6-s − 0.999·8-s + (1 + 1.73i)9-s + (0.5 − 0.866i)11-s + (0.499 + 0.866i)12-s + 13-s + (−0.5 − 0.866i)16-s + (−3 + 5.19i)17-s + (−1 + 1.73i)18-s + (1 + 1.73i)19-s + 0.999·22-s + (3 + 5.19i)23-s + (−0.499 + 0.866i)24-s + ⋯ |
L(s) = 1 | + (0.353 + 0.612i)2-s + (0.288 − 0.499i)3-s + (−0.249 + 0.433i)4-s + 0.408·6-s − 0.353·8-s + (0.333 + 0.577i)9-s + (0.150 − 0.261i)11-s + (0.144 + 0.249i)12-s + 0.277·13-s + (−0.125 − 0.216i)16-s + (−0.727 + 1.26i)17-s + (−0.235 + 0.408i)18-s + (0.229 + 0.397i)19-s + 0.213·22-s + (0.625 + 1.08i)23-s + (−0.102 + 0.176i)24-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1078 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.386 - 0.922i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.127838218\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.127838218\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.5 - 0.866i)T \) |
| 7 | \( 1 \) |
| 11 | \( 1 + (-0.5 + 0.866i)T \) |
good | 3 | \( 1 + (-0.5 + 0.866i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (-2.5 + 4.33i)T^{2} \) |
| 13 | \( 1 - T + 13T^{2} \) |
| 17 | \( 1 + (3 - 5.19i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1 - 1.73i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-3 - 5.19i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 9T + 29T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1 + 1.73i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + (3 + 5.19i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (1.5 - 2.59i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.5 - 9.52i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (5.5 - 9.52i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + (-1 + 1.73i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (2.5 + 4.33i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6T + 83T^{2} \) |
| 89 | \( 1 + (9 + 15.5i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 13T + 97T^{2} \) |
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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
−10.08415165423804370100830699077, −8.748428681230517740595970847040, −8.408241477529121046090652651866, −7.44747866639599614575568703753, −6.72806210687619591578597191674, −5.91832664086939168897531281991, −4.88251649371443125774688056225, −3.96281355148462004812862051896, −2.79874522381940100093547824940, −1.45698573322672188167741303031,
0.926571860457696678279022541406, 2.54536368458609811717449923553, 3.37129338803553429510955927136, 4.49004701694734968738406571790, 4.98380498282920744647575634358, 6.40874128885435513045069454441, 7.01948105860642191046275300789, 8.347297881760747658864483890905, 9.255661462487002235834745161795, 9.587159948076271020332804923471