L(s) = 1 | + (−75.2 + 75.2i)3-s + (−14.1 + 624. i)5-s + (−730. − 730. i)7-s − 4.77e3i·9-s − 1.95e4·11-s + (−2.49e4 + 2.49e4i)13-s + (−4.59e4 − 4.81e4i)15-s + (−1.12e4 − 1.12e4i)17-s + 1.71e5i·19-s + 1.10e5·21-s + (1.32e5 − 1.32e5i)23-s + (−3.90e5 − 1.77e4i)25-s + (−1.34e5 − 1.34e5i)27-s − 1.27e5i·29-s + 9.60e5·31-s + ⋯ |
L(s) = 1 | + (−0.929 + 0.929i)3-s + (−0.0226 + 0.999i)5-s + (−0.304 − 0.304i)7-s − 0.728i·9-s − 1.33·11-s + (−0.872 + 0.872i)13-s + (−0.908 − 0.950i)15-s + (−0.135 − 0.135i)17-s + 1.31i·19-s + 0.566·21-s + (0.471 − 0.471i)23-s + (−0.998 − 0.0453i)25-s + (−0.252 − 0.252i)27-s − 0.179i·29-s + 1.04·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.506 + 0.862i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(0.0505181 - 0.0289212i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0505181 - 0.0289212i\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (14.1 - 624. i)T \) |
good | 3 | \( 1 + (75.2 - 75.2i)T - 6.56e3iT^{2} \) |
| 7 | \( 1 + (730. + 730. i)T + 5.76e6iT^{2} \) |
| 11 | \( 1 + 1.95e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (2.49e4 - 2.49e4i)T - 8.15e8iT^{2} \) |
| 17 | \( 1 + (1.12e4 + 1.12e4i)T + 6.97e9iT^{2} \) |
| 19 | \( 1 - 1.71e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + (-1.32e5 + 1.32e5i)T - 7.83e10iT^{2} \) |
| 29 | \( 1 + 1.27e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 9.60e5T + 8.52e11T^{2} \) |
| 37 | \( 1 + (2.43e5 + 2.43e5i)T + 3.51e12iT^{2} \) |
| 41 | \( 1 - 2.50e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + (-6.76e3 + 6.76e3i)T - 1.16e13iT^{2} \) |
| 47 | \( 1 + (-1.79e6 - 1.79e6i)T + 2.38e13iT^{2} \) |
| 53 | \( 1 + (2.97e6 - 2.97e6i)T - 6.22e13iT^{2} \) |
| 59 | \( 1 + 3.13e5iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.76e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + (4.41e6 + 4.41e6i)T + 4.06e14iT^{2} \) |
| 71 | \( 1 - 8.89e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (-1.95e7 + 1.95e7i)T - 8.06e14iT^{2} \) |
| 79 | \( 1 - 1.11e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + (1.58e7 - 1.58e7i)T - 2.25e15iT^{2} \) |
| 89 | \( 1 - 4.85e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 + (1.07e8 + 1.07e8i)T + 7.83e15iT^{2} \) |
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
−12.30508358540888870546679333240, −11.18845661260771095508861246027, −10.37552123030360057212856919312, −9.750655759135497441925957573801, −7.79583394226201046953159953679, −6.54553505705720774132572717519, −5.31963634276963115047610183701, −4.09517441880961562353551136064, −2.53805152763485066221945326126, −0.02626370864021809026734859977,
0.867108337909376549697093389960, 2.57958159985600960044881916329, 4.92613627391554804561551871827, 5.68699365601840398591832796249, 7.11244887038936403794091307687, 8.179971708402015616785197619103, 9.584387931677775788583162812422, 10.94834952883202483137736285136, 12.10232686862502035264521754992, 12.84126671299566647855113493687