| L(s) = 1 | + 1.41i·2-s + (−2.64 + 1.41i)3-s − 2.00·4-s + 6.57i·5-s + (−2.00 − 3.74i)6-s + 2.64·7-s − 2.82i·8-s + (5 − 7.48i)9-s − 9.29·10-s + 0.412i·11-s + (5.29 − 2.82i)12-s + 20.5·13-s + 3.74i·14-s + (−9.29 − 17.3i)15-s + 4.00·16-s + 15.8i·17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (−0.881 + 0.471i)3-s − 0.500·4-s + 1.31i·5-s + (−0.333 − 0.623i)6-s + 0.377·7-s − 0.353i·8-s + (0.555 − 0.831i)9-s − 0.929·10-s + 0.0374i·11-s + (0.440 − 0.235i)12-s + 1.58·13-s + 0.267i·14-s + (−0.619 − 1.15i)15-s + 0.250·16-s + 0.934i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 42 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.471 - 0.881i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.428707 + 0.715262i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.428707 + 0.715262i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (2.64 - 1.41i)T \) |
| 7 | \( 1 - 2.64T \) |
| good | 5 | \( 1 - 6.57iT - 25T^{2} \) |
| 11 | \( 1 - 0.412iT - 121T^{2} \) |
| 13 | \( 1 - 20.5T + 169T^{2} \) |
| 17 | \( 1 - 15.8iT - 289T^{2} \) |
| 19 | \( 1 + 16T + 361T^{2} \) |
| 23 | \( 1 + 36.0iT - 529T^{2} \) |
| 29 | \( 1 - 20.8iT - 841T^{2} \) |
| 31 | \( 1 + 5.54T + 961T^{2} \) |
| 37 | \( 1 - 20T + 1.36e3T^{2} \) |
| 41 | \( 1 + 76.1iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 51.7T + 1.84e3T^{2} \) |
| 47 | \( 1 + 8.48iT - 2.20e3T^{2} \) |
| 53 | \( 1 + 50.9iT - 2.80e3T^{2} \) |
| 59 | \( 1 + 1.64iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 66.9T + 3.72e3T^{2} \) |
| 67 | \( 1 - 49.4T + 4.48e3T^{2} \) |
| 71 | \( 1 - 87.7iT - 5.04e3T^{2} \) |
| 73 | \( 1 + 12.3T + 5.32e3T^{2} \) |
| 79 | \( 1 + 84.9T + 6.24e3T^{2} \) |
| 83 | \( 1 - 4.12iT - 6.88e3T^{2} \) |
| 89 | \( 1 - 31.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 68.8T + 9.40e3T^{2} \) |
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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
−16.10563307311052372998004308932, −15.12020900953462530939239146992, −14.28805538319360899718830478820, −12.69250280727457395850480271300, −10.99922140489800898182747560996, −10.48806283517269266828015213592, −8.618024251648609891486181622236, −6.82977106379627389499134696531, −5.93492721119400911993870894085, −4.00088304407135574630311318648,
1.23197583297330044693123404601, 4.49681830592379270655001915085, 5.84403132523823845965565988871, 7.984828802260199342245516498849, 9.297629870369767403113554207709, 10.97656607736838537880749855765, 11.80190145350659534624340353683, 12.96017405790387703127267444159, 13.63441889211531503309611650261, 15.75622416139650351066501500768
