| L(s) = 1 | + 14.9i·2-s + 32.7·4-s + 1.08e3i·5-s + 1.30e3·7-s + 4.31e3i·8-s − 1.62e4·10-s + 1.41e4i·11-s + 2.83e4·13-s + 1.95e4i·14-s − 5.60e4·16-s + 9.91e3i·17-s + 2.05e5·19-s + 3.56e4i·20-s − 2.11e5·22-s − 1.30e5i·23-s + ⋯ |
| L(s) = 1 | + 0.933i·2-s + 0.127·4-s + 1.74i·5-s + 0.544·7-s + 1.05i·8-s − 1.62·10-s + 0.967i·11-s + 0.993·13-s + 0.508i·14-s − 0.855·16-s + 0.118i·17-s + 1.57·19-s + 0.222i·20-s − 0.903·22-s − 0.467i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.53043i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.53043i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 14.9iT - 256T^{2} \) |
| 5 | \( 1 - 1.08e3iT - 3.90e5T^{2} \) |
| 7 | \( 1 - 1.30e3T + 5.76e6T^{2} \) |
| 11 | \( 1 - 1.41e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.83e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 9.91e3iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 2.05e5T + 1.69e10T^{2} \) |
| 23 | \( 1 + 1.30e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 6.94e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 - 1.33e6T + 8.52e11T^{2} \) |
| 37 | \( 1 + 2.71e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 5.05e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 3.08e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + 3.86e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 5.11e6iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 1.72e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 9.54e6T + 1.91e14T^{2} \) |
| 67 | \( 1 - 2.03e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.46e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 4.84e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.99e7T + 1.51e15T^{2} \) |
| 83 | \( 1 - 3.61e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.92e7iT - 3.93e15T^{2} \) |
| 97 | \( 1 - 1.50e8T + 7.83e15T^{2} \) |
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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
−13.85779685923147812389347793042, −11.90735007370282123502638443320, −11.03604868099338046572643030103, −10.10816509481395776369856120446, −8.328771320106349428609140832928, −7.18950293293362015684834455528, −6.63268555485182995212066953992, −5.27714786580019163225080258841, −3.33581742422237550119689651963, −1.94992118814861303424287802970,
0.829107139999178940130430003281, 1.44029283670533361303835287018, 3.30175609313197157169790942072, 4.67952883342745729490467052578, 5.97255211027569631300287125194, 7.916712894741322437456039479993, 8.898449267328743887797393512954, 9.992927956257851928803215436146, 11.48758449043425305183465880388, 11.84414210157417888948065979545
