| L(s) = 1 | + (2.23 − 2i)3-s + 4·5-s + 8.94·7-s + (1.00 − 8.94i)9-s − 4.47·11-s − 17.8i·13-s + (8.94 − 8i)15-s + 17.8i·17-s + 20i·19-s + (20.0 − 17.8i)21-s + 16i·23-s − 9·25-s + (−15.6 − 22.0i)27-s + 52·29-s − 26.8·31-s + ⋯ |
| L(s) = 1 | + (0.745 − 0.666i)3-s + 0.800·5-s + 1.27·7-s + (0.111 − 0.993i)9-s − 0.406·11-s − 1.37i·13-s + (0.596 − 0.533i)15-s + 1.05i·17-s + 1.05i·19-s + (0.952 − 0.851i)21-s + 0.695i·23-s − 0.359·25-s + (−0.579 − 0.814i)27-s + 1.79·29-s − 0.865·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.666 + 0.745i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(2.49674 - 1.11657i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.49674 - 1.11657i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-2.23 + 2i)T \) |
| good | 5 | \( 1 - 4T + 25T^{2} \) |
| 7 | \( 1 - 8.94T + 49T^{2} \) |
| 11 | \( 1 + 4.47T + 121T^{2} \) |
| 13 | \( 1 + 17.8iT - 169T^{2} \) |
| 17 | \( 1 - 17.8iT - 289T^{2} \) |
| 19 | \( 1 - 20iT - 361T^{2} \) |
| 23 | \( 1 - 16iT - 529T^{2} \) |
| 29 | \( 1 - 52T + 841T^{2} \) |
| 31 | \( 1 + 26.8T + 961T^{2} \) |
| 37 | \( 1 + 53.6iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 35.7iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 36iT - 1.84e3T^{2} \) |
| 47 | \( 1 - 64iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 20T + 2.80e3T^{2} \) |
| 59 | \( 1 + 102.T + 3.48e3T^{2} \) |
| 61 | \( 1 + 17.8iT - 3.72e3T^{2} \) |
| 67 | \( 1 - 44iT - 4.48e3T^{2} \) |
| 71 | \( 1 - 80iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 50T + 5.32e3T^{2} \) |
| 79 | \( 1 - 80.4T + 6.24e3T^{2} \) |
| 83 | \( 1 - 102.T + 6.88e3T^{2} \) |
| 89 | \( 1 - 160. iT - 7.92e3T^{2} \) |
| 97 | \( 1 - 50T + 9.40e3T^{2} \) |
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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
−10.82409359977997777896225469463, −10.15600259984593330417874252276, −8.977758037471863768761610954872, −8.045928238554707265765580298313, −7.62384514662258914622161248502, −6.11202549994904674914148791954, −5.34167209149257630168974521349, −3.76308964778912732291966214090, −2.34673273055620304021851846479, −1.33722842425845759801523838001,
1.78058592660570432091408805083, 2.81650044889059464762199055879, 4.60210958180190362129053069945, 4.93895365919870181489582044968, 6.53362004306384823118236054964, 7.69914222225944950952510561439, 8.653175702545908227729403741552, 9.349966056313222400939338187645, 10.22850531829132248946279740521, 11.14903888108388698909715004678
