L(s) = 1 | + (3.90 + 0.888i)2-s + (3.67 − 3.67i)3-s + (14.4 + 6.92i)4-s + (−17.3 + 17.3i)5-s + (17.5 − 11.0i)6-s + 89.8·7-s + (50.0 + 39.8i)8-s − 27i·9-s + (−83.1 + 52.3i)10-s + (−94.2 − 94.2i)11-s + (78.4 − 27.5i)12-s + (−100. − 100. i)13-s + (350. + 79.7i)14-s + 127. i·15-s + (160. + 199. i)16-s − 66.1·17-s + ⋯ |
L(s) = 1 | + (0.975 + 0.222i)2-s + (0.408 − 0.408i)3-s + (0.901 + 0.432i)4-s + (−0.694 + 0.694i)5-s + (0.488 − 0.307i)6-s + 1.83·7-s + (0.782 + 0.622i)8-s − 0.333i·9-s + (−0.831 + 0.523i)10-s + (−0.779 − 0.779i)11-s + (0.544 − 0.191i)12-s + (−0.596 − 0.596i)13-s + (1.78 + 0.407i)14-s + 0.567i·15-s + (0.625 + 0.780i)16-s − 0.228·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 48 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.960 - 0.278i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{5}{2})\) |
\(\approx\) |
\(2.89745 + 0.412045i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.89745 + 0.412045i\) |
\(L(3)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-3.90 - 0.888i)T \) |
| 3 | \( 1 + (-3.67 + 3.67i)T \) |
good | 5 | \( 1 + (17.3 - 17.3i)T - 625iT^{2} \) |
| 7 | \( 1 - 89.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + (94.2 + 94.2i)T + 1.46e4iT^{2} \) |
| 13 | \( 1 + (100. + 100. i)T + 2.85e4iT^{2} \) |
| 17 | \( 1 + 66.1T + 8.35e4T^{2} \) |
| 19 | \( 1 + (324. - 324. i)T - 1.30e5iT^{2} \) |
| 23 | \( 1 + 467.T + 2.79e5T^{2} \) |
| 29 | \( 1 + (321. + 321. i)T + 7.07e5iT^{2} \) |
| 31 | \( 1 + 1.58e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + (-982. + 982. i)T - 1.87e6iT^{2} \) |
| 41 | \( 1 + 1.21e3iT - 2.82e6T^{2} \) |
| 43 | \( 1 + (-1.30e3 - 1.30e3i)T + 3.41e6iT^{2} \) |
| 47 | \( 1 - 3.97e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + (-672. + 672. i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 + (1.18e3 + 1.18e3i)T + 1.21e7iT^{2} \) |
| 61 | \( 1 + (979. + 979. i)T + 1.38e7iT^{2} \) |
| 67 | \( 1 + (2.26e3 - 2.26e3i)T - 2.01e7iT^{2} \) |
| 71 | \( 1 - 5.87e3T + 2.54e7T^{2} \) |
| 73 | \( 1 - 3.66e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 539. iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (3.60e3 - 3.60e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 7.54e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + 9.59e3T + 8.85e7T^{2} \) |
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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
−14.75034477865395554324242168921, −14.10440419691923239329087748771, −12.77102068555833428049933948028, −11.50334146848028200388130747796, −10.80318888548558341200684613161, −7.962150603521547065725421800790, −7.74465208087770285652743420846, −5.77724068340738282159196527374, −4.14051611204220703200694136361, −2.35528311257401121496858027811,
2.05914216131395043421815910228, 4.42800438555064968161042700815, 4.93199223812630055316330831767, 7.39290151720046191796674707182, 8.541079690013912726685138264219, 10.43622099395609515708893432739, 11.54141613578511619557287965978, 12.47737207383960717693781866765, 13.86563931598327716553801599793, 14.86306879984909246133420360023