| L(s) = 1 | + 7.44·2-s + (37.3 + 64.6i)3-s − 72.5·4-s + (−250. − 434. i)5-s + (278. + 481. i)6-s + (−47.3 + 82.0i)7-s − 1.49e3·8-s + (−1.69e3 + 2.93e3i)9-s + (−1.86e3 − 3.23e3i)10-s − 4.86e3·11-s + (−2.70e3 − 4.69e3i)12-s + (−3.38e3 + 5.86e3i)13-s + (−352. + 611. i)14-s + (1.87e4 − 3.24e4i)15-s − 1.84e3·16-s + (9.83e3 − 1.70e4i)17-s + ⋯ |
| L(s) = 1 | + 0.658·2-s + (0.798 + 1.38i)3-s − 0.566·4-s + (−0.897 − 1.55i)5-s + (0.525 + 0.910i)6-s + (−0.0522 + 0.0904i)7-s − 1.03·8-s + (−0.775 + 1.34i)9-s + (−0.591 − 1.02i)10-s − 1.10·11-s + (−0.452 − 0.783i)12-s + (−0.427 + 0.739i)13-s + (−0.0343 + 0.0595i)14-s + (1.43 − 2.48i)15-s − 0.112·16-s + (0.485 − 0.840i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 43 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.860 + 0.509i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.00577667 - 0.0210941i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.00577667 - 0.0210941i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 43 | \( 1 + (5.10e5 + 1.07e5i)T \) |
| good | 2 | \( 1 - 7.44T + 128T^{2} \) |
| 3 | \( 1 + (-37.3 - 64.6i)T + (-1.09e3 + 1.89e3i)T^{2} \) |
| 5 | \( 1 + (250. + 434. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (47.3 - 82.0i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + 4.86e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (3.38e3 - 5.86e3i)T + (-3.13e7 - 5.43e7i)T^{2} \) |
| 17 | \( 1 + (-9.83e3 + 1.70e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (4.70e3 + 8.14e3i)T + (-4.46e8 + 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.20e4 + 5.55e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (5.06e4 - 8.77e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-8.17e4 - 1.41e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.11e5 - 3.66e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 4.53e5T + 1.94e11T^{2} \) |
| 47 | \( 1 + 2.47e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + (8.41e5 + 1.45e6i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + 2.55e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + (-1.46e5 + 2.54e5i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (2.09e6 + 3.63e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (2.58e6 - 4.48e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (-2.80e6 + 4.85e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (1.65e6 - 2.87e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (1.00e6 + 1.74e6i)T + (-1.35e13 + 2.35e13i)T^{2} \) |
| 89 | \( 1 + (3.52e6 + 6.09e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.46e7T + 8.07e13T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−15.18090432897786351753846717685, −14.05758819794977016117011169327, −12.92447151124414129370604019158, −11.84854773953499893155937616967, −9.951946288622130184149615602509, −8.943472087747184099188930646317, −8.182503887816589886587409188912, −4.95119019267006297352017104670, −4.64181856198908995607030312549, −3.25460120686881800382762931260,
0.00650701639483671922559670019, 2.59620589133635263093524397981, 3.62411482248869156790193330019, 6.01870969923017568172152055593, 7.52396056800803537486352347237, 8.096523986119547718938533668556, 10.18478065260044991938769153578, 11.79620656645858388128725685834, 12.87858808807653899902811693737, 13.71203378952664935646046050009
