| L(s) = 1 | + (4.38 − 2.52i)2-s + (−51.2 + 88.6i)4-s + (258. + 447. i)5-s + (−900. − 111. i)7-s + 1.16e3i·8-s + (2.26e3 + 1.30e3i)10-s + (−4.61e3 − 2.66e3i)11-s − 5.26e3i·13-s + (−4.22e3 + 1.78e3i)14-s + (−3.60e3 − 6.24e3i)16-s + (1.24e4 − 2.14e4i)17-s + (−1.31e4 + 7.58e3i)19-s − 5.28e4·20-s − 2.69e4·22-s + (−3.34e4 + 1.93e4i)23-s + ⋯ |
| L(s) = 1 | + (0.387 − 0.223i)2-s + (−0.400 + 0.692i)4-s + (0.923 + 1.60i)5-s + (−0.992 − 0.122i)7-s + 0.804i·8-s + (0.715 + 0.413i)10-s + (−1.04 − 0.604i)11-s − 0.664i·13-s + (−0.411 + 0.174i)14-s + (−0.220 − 0.381i)16-s + (0.612 − 1.06i)17-s + (−0.439 + 0.253i)19-s − 1.47·20-s − 0.540·22-s + (−0.573 + 0.330i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0882i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.996 - 0.0882i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.0407364 + 0.921221i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0407364 + 0.921221i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (900. + 111. i)T \) |
| good | 2 | \( 1 + (-4.38 + 2.52i)T + (64 - 110. i)T^{2} \) |
| 5 | \( 1 + (-258. - 447. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 11 | \( 1 + (4.61e3 + 2.66e3i)T + (9.74e6 + 1.68e7i)T^{2} \) |
| 13 | \( 1 + 5.26e3iT - 6.27e7T^{2} \) |
| 17 | \( 1 + (-1.24e4 + 2.14e4i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 19 | \( 1 + (1.31e4 - 7.58e3i)T + (4.46e8 - 7.74e8i)T^{2} \) |
| 23 | \( 1 + (3.34e4 - 1.93e4i)T + (1.70e9 - 2.94e9i)T^{2} \) |
| 29 | \( 1 - 1.86e4iT - 1.72e10T^{2} \) |
| 31 | \( 1 + (-1.43e5 - 8.29e4i)T + (1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + (-2.32e5 - 4.03e5i)T + (-4.74e10 + 8.22e10i)T^{2} \) |
| 41 | \( 1 + 4.66e5T + 1.94e11T^{2} \) |
| 43 | \( 1 + 8.92e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + (-3.95e4 - 6.84e4i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (1.81e5 + 1.04e5i)T + (5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (1.11e6 - 1.93e6i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (-2.06e6 + 1.19e6i)T + (1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.12e6 - 1.95e6i)T + (-3.03e12 - 5.24e12i)T^{2} \) |
| 71 | \( 1 - 1.19e5iT - 9.09e12T^{2} \) |
| 73 | \( 1 + (-8.45e4 - 4.88e4i)T + (5.52e12 + 9.56e12i)T^{2} \) |
| 79 | \( 1 + (7.02e5 + 1.21e6i)T + (-9.60e12 + 1.66e13i)T^{2} \) |
| 83 | \( 1 - 1.04e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (7.51e5 + 1.30e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 - 1.54e7iT - 8.07e13T^{2} \) |
| 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
−13.62364799057132313523127814898, −13.35684939346327433210772123849, −11.86017203574326495767340869043, −10.51824204601392542675195861907, −9.771512117941808376691408170620, −8.013992310051067077284238882992, −6.71168820724203040840662061446, −5.45239534988942456752245231339, −3.26192391977792640937406494267, −2.73795764606513754916118380939,
0.27837172180552514955990269069, 1.84753978555022263172973002523, 4.32760514685581702686362761671, 5.41929673630405794981648448073, 6.34841484300804819471960030620, 8.450032134327731704270978982132, 9.608189584789226492628475380898, 10.16392755248985405798963039857, 12.43750597392742342827739224851, 13.02565959179878459444200362476
