| L(s) = 1 | + (−6.57 + 6.14i)3-s + 9.58i·5-s − 22.6·7-s + (5.37 − 80.8i)9-s − 2.72i·11-s + 220.·13-s + (−58.9 − 63.0i)15-s + 172. i·17-s + 275.·19-s + (149. − 139. i)21-s + 420. i·23-s + 533.·25-s + (461. + 564. i)27-s − 207. i·29-s − 1.08e3·31-s + ⋯ |
| L(s) = 1 | + (−0.730 + 0.683i)3-s + 0.383i·5-s − 0.462·7-s + (0.0663 − 0.997i)9-s − 0.0224i·11-s + 1.30·13-s + (−0.262 − 0.280i)15-s + 0.598i·17-s + 0.763·19-s + (0.337 − 0.316i)21-s + 0.794i·23-s + 0.852·25-s + (0.633 + 0.773i)27-s − 0.246i·29-s − 1.13·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.683 - 0.730i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.110593113\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.110593113\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (6.57 - 6.14i)T \) |
| good | 5 | \( 1 - 9.58iT - 625T^{2} \) |
| 7 | \( 1 + 22.6T + 2.40e3T^{2} \) |
| 11 | \( 1 + 2.72iT - 1.46e4T^{2} \) |
| 13 | \( 1 - 220.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 172. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 275.T + 1.30e5T^{2} \) |
| 23 | \( 1 - 420. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 207. iT - 7.07e5T^{2} \) |
| 31 | \( 1 + 1.08e3T + 9.23e5T^{2} \) |
| 37 | \( 1 - 1.26e3T + 1.87e6T^{2} \) |
| 41 | \( 1 - 1.23e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + 1.38e3T + 3.41e6T^{2} \) |
| 47 | \( 1 + 1.43e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 3.63e3iT - 7.89e6T^{2} \) |
| 59 | \( 1 - 5.98e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 - 2.17e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 5.98e3T + 2.01e7T^{2} \) |
| 71 | \( 1 - 7.55e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 3.32e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 279.T + 3.89e7T^{2} \) |
| 83 | \( 1 - 4.33e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.25e4iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 1.65e4T + 8.85e7T^{2} \) |
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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
−11.08409606214666787538615707269, −10.20754584051924784311610726890, −9.423681015780541119131441324613, −8.429755589627534145970432629041, −7.05433269496925547554566089448, −6.16909091581940241471432843620, −5.36477827484755420966994683259, −4.00121131342934720892996813837, −3.18491860796397735404699267995, −1.18547075011575867314507262840,
0.40691974046251598879188800535, 1.51110513334652422859298025936, 3.10856344630003466937329984382, 4.60325329534734146881625650612, 5.65759300471766602869932703552, 6.50043901703182486837402097637, 7.42134656482815300527928818716, 8.479858466753080267527807532397, 9.414806381878163260559457644514, 10.63099978610008457570290269861
