L(s) = 1 | + 4.86i·2-s − 15.6·4-s − 12.7·5-s − 37.3i·8-s − 62.1i·10-s − 54.1i·11-s − 8.85i·13-s + 56.2·16-s + 68.9·17-s + 163. i·19-s + 200.·20-s + 263.·22-s − 93.9i·23-s + 37.9·25-s + 43.0·26-s + ⋯ |
L(s) = 1 | + 1.72i·2-s − 1.95·4-s − 1.14·5-s − 1.64i·8-s − 1.96i·10-s − 1.48i·11-s − 0.188i·13-s + 0.878·16-s + 0.983·17-s + 1.97i·19-s + 2.23·20-s + 2.55·22-s − 0.851i·23-s + 0.303·25-s + 0.324·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (-0.0980 - 0.995i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(2)\) |
\(\approx\) |
\(1.114579802\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.114579802\) |
\(L(\frac{5}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 - 4.86iT - 8T^{2} \) |
| 5 | \( 1 + 12.7T + 125T^{2} \) |
| 11 | \( 1 + 54.1iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 8.85iT - 2.19e3T^{2} \) |
| 17 | \( 1 - 68.9T + 4.91e3T^{2} \) |
| 19 | \( 1 - 163. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 93.9iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 119. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 98.8iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 94.1T + 5.06e4T^{2} \) |
| 41 | \( 1 - 259.T + 6.89e4T^{2} \) |
| 43 | \( 1 - 5.01T + 7.95e4T^{2} \) |
| 47 | \( 1 + 57.3T + 1.03e5T^{2} \) |
| 53 | \( 1 - 470. iT - 1.48e5T^{2} \) |
| 59 | \( 1 - 225.T + 2.05e5T^{2} \) |
| 61 | \( 1 + 427. iT - 2.26e5T^{2} \) |
| 67 | \( 1 - 163.T + 3.00e5T^{2} \) |
| 71 | \( 1 + 79.8iT - 3.57e5T^{2} \) |
| 73 | \( 1 - 769. iT - 3.89e5T^{2} \) |
| 79 | \( 1 - 534.T + 4.93e5T^{2} \) |
| 83 | \( 1 - 438.T + 5.71e5T^{2} \) |
| 89 | \( 1 + 25.6T + 7.04e5T^{2} \) |
| 97 | \( 1 + 1.38e3iT - 9.12e5T^{2} \) |
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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
−10.91324911521582694163040753505, −9.766119825317747454018489323976, −8.481585550364094438085493114006, −8.111011303489316117273833126073, −7.42867145227371881667640538578, −6.17087551840074081569534425244, −5.62779577992597780008613305705, −4.29240059027393556410520260538, −3.40331653230496979546404011331, −0.60362025812336524623301746553,
0.75182704675420247217558092807, 2.17061358834355910707038972717, 3.32886645065176058336533053958, 4.27188497622941433070188493108, 5.04206583619785448342283122815, 7.01314521432208620747918887033, 7.84242692011508809869052621292, 9.108681528805160981960803305199, 9.695267157672675149633665253066, 10.68487891882800203542553623200