L(s) = 1 | + 16.9i·3-s + (−12.3 + 54.5i)5-s + 211. i·7-s − 45.4·9-s − 520.·11-s − 732. i·13-s + (−925. − 210. i)15-s + 2.26e3i·17-s − 2.03e3·19-s − 3.59e3·21-s − 974. i·23-s + (−2.81e3 − 1.35e3i)25-s + 3.35e3i·27-s + 5.27e3·29-s + 2.00e3·31-s + ⋯ |
L(s) = 1 | + 1.08i·3-s + (−0.221 + 0.975i)5-s + 1.63i·7-s − 0.186·9-s − 1.29·11-s − 1.20i·13-s + (−1.06 − 0.241i)15-s + 1.90i·17-s − 1.29·19-s − 1.77·21-s − 0.384i·23-s + (−0.901 − 0.432i)25-s + 0.885i·27-s + 1.16·29-s + 0.374·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.221 + 0.975i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(3)\) |
\(\approx\) |
\(0.9511202659\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.9511202659\) |
\(L(\frac{7}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (12.3 - 54.5i)T \) |
good | 3 | \( 1 - 16.9iT - 243T^{2} \) |
| 7 | \( 1 - 211. iT - 1.68e4T^{2} \) |
| 11 | \( 1 + 520.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 732. iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 2.26e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.03e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 974. iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 5.27e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.00e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 3.65e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.71e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 4.07e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.33e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 2.74e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 3.89e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 8.73e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 4.09e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.68e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 9.82e3iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 6.03e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 2.41e3iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 6.93e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + 7.62e4iT - 8.58e9T^{2} \) |
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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
−11.05439212969052698194308020976, −10.51577552502882959122763932622, −9.902446485915983567669342475572, −8.533323888443356762728647750248, −8.017809361871125398074701201173, −6.32584951141547630228438464067, −5.58987183067402318391613669617, −4.42341173762134746929482278571, −3.13540065974709898775412399671, −2.32520833178898893449799360115,
0.28139434845472859232736966640, 1.03251431655905680565838913091, 2.34844046212926818003852036256, 4.16343149945734517591673071165, 4.90805109768856136321911537507, 6.50661609001907851788120241212, 7.37989083323089430877274140439, 7.87622180558621879953545509586, 9.095915270628078094285124757756, 10.17098621156149274036331725972