| L(s) = 1 | + (−1.14 − 15.5i)3-s − 71.6·5-s − 100. i·7-s + (−240. + 35.5i)9-s + 148. i·11-s − 381. i·13-s + (82.0 + 1.11e3i)15-s − 2.20e3i·17-s − 1.73e3·19-s + (−1.56e3 + 114. i)21-s − 3.55e3·23-s + 2.01e3·25-s + (827. + 3.69e3i)27-s − 5.77e3·29-s − 5.12e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.0733 − 0.997i)3-s − 1.28·5-s − 0.775i·7-s + (−0.989 + 0.146i)9-s + 0.369i·11-s − 0.625i·13-s + (0.0941 + 1.27i)15-s − 1.85i·17-s − 1.10·19-s + (−0.773 + 0.0568i)21-s − 1.40·23-s + 0.644·25-s + (0.218 + 0.975i)27-s − 1.27·29-s − 0.957i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.653 - 0.757i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.04182534600\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.04182534600\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.14 + 15.5i)T \) |
| good | 5 | \( 1 + 71.6T + 3.12e3T^{2} \) |
| 7 | \( 1 + 100. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 148. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + 381. iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 2.20e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 1.73e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 3.55e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 5.77e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 5.12e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 1.10e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.45e3iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 1.86e4T + 1.47e8T^{2} \) |
| 47 | \( 1 + 1.24e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 4.84e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 1.82e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 5.45e3iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 2.22e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 1.03e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 6.12e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 3.13e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 4.66e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 9.97e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 - 1.15e5T + 8.58e9T^{2} \) |
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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
−10.95084803057479475405628921217, −9.748537114670822453621547670838, −8.455832064389288684403944458704, −7.59484600127313641738851141509, −7.25798913221808505496296600680, −6.03782635254364559326801106680, −4.66368970944739662984306429681, −3.61251251476960021927969831341, −2.29850567977790189165467261360, −0.68968983054333569520580031617,
0.01605871379762542718639169215, 2.16213079050760838367347209744, 3.80178293410221930020368081777, 4.05666074210324608701990088937, 5.50422427648061346186635319556, 6.39206429140998549654623977409, 7.953601034200250956294059441134, 8.532004510069046997805628070498, 9.378470647878836792386535936416, 10.63801002119734056107483223387
