| L(s) = 1 | − 14.4i·2-s − 49.3i·3-s + 47.3·4-s + 738. i·5-s − 712.·6-s − 198. i·7-s − 4.38e3i·8-s + 4.12e3·9-s + 1.06e4·10-s + 1.24e4·11-s − 2.33e3i·12-s + 2.78e4·13-s − 2.86e3·14-s + 3.64e4·15-s − 5.11e4·16-s − 6.64e4·17-s + ⋯ |
| L(s) = 1 | − 0.902i·2-s − 0.609i·3-s + 0.184·4-s + 1.18i·5-s − 0.549·6-s − 0.0826i·7-s − 1.06i·8-s + 0.629·9-s + 1.06·10-s + 0.849·11-s − 0.112i·12-s + 0.974·13-s − 0.0746·14-s + 0.719·15-s − 0.781·16-s − 0.795·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.0437 + 0.999i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (-0.0437 + 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.65695 - 1.73106i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.65695 - 1.73106i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (1.49e5 - 3.41e6i)T \) |
| good | 2 | \( 1 + 14.4iT - 256T^{2} \) |
| 3 | \( 1 + 49.3iT - 6.56e3T^{2} \) |
| 5 | \( 1 - 738. iT - 3.90e5T^{2} \) |
| 7 | \( 1 + 198. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.24e4T + 2.14e8T^{2} \) |
| 13 | \( 1 - 2.78e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 6.64e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.54e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 2.53e5T + 7.83e10T^{2} \) |
| 29 | \( 1 - 3.32e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + 7.16e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.30e6iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 1.26e5T + 7.98e12T^{2} \) |
| 47 | \( 1 - 3.57e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 4.88e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 5.89e6T + 1.46e14T^{2} \) |
| 61 | \( 1 - 3.89e6iT - 1.91e14T^{2} \) |
| 67 | \( 1 - 2.99e7T + 4.06e14T^{2} \) |
| 71 | \( 1 + 4.99e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 1.46e7iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 1.87e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 7.06e7T + 2.25e15T^{2} \) |
| 89 | \( 1 - 1.63e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + 1.07e8T + 7.83e15T^{2} \) |
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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
−13.59724178135724369422502704966, −12.63572078557354469218888593495, −11.26240979429042131263574358657, −10.75923375755566151870536236950, −9.237334428137844785872540402609, −7.12371833633735312736146124275, −6.56711375415879341471364632252, −3.84555748014152724487315649467, −2.46002686138896517816116517553, −1.08888187820332121352924683995,
1.44396101608817631505358528992, 4.06258067852579040687085557775, 5.37875039537024956748854667721, 6.74617695005928348121407227318, 8.326750103906845158650565845145, 9.262294687585012302136681825716, 10.87444899833908303772257813982, 12.20432180730008424662693537295, 13.50564395533573427527470657893, 14.91216758210998081367655266451
