| L(s) = 1 | + (51.3 − 62.6i)3-s + (918. − 530. i)5-s + (−1.25e3 + 2.04e3i)7-s + (−1.29e3 − 6.43e3i)9-s + (2.20e4 + 1.27e4i)11-s + 4.82e4·13-s + (1.39e4 − 8.48e4i)15-s + (1.51e4 + 8.72e3i)17-s + (6.76e4 + 1.17e5i)19-s + (6.37e4 + 1.83e5i)21-s + (−4.28e5 + 2.47e5i)23-s + (3.67e5 − 6.36e5i)25-s + (−4.69e5 − 2.49e5i)27-s − 4.48e5i·29-s + (1.92e5 − 3.32e5i)31-s + ⋯ |
| L(s) = 1 | + (0.633 − 0.773i)3-s + (1.47 − 0.848i)5-s + (−0.523 + 0.852i)7-s + (−0.196 − 0.980i)9-s + (1.50 + 0.870i)11-s + 1.69·13-s + (0.275 − 1.67i)15-s + (0.180 + 0.104i)17-s + (0.519 + 0.899i)19-s + (0.327 + 0.944i)21-s + (−1.52 + 0.883i)23-s + (0.941 − 1.63i)25-s + (−0.883 − 0.469i)27-s − 0.633i·29-s + (0.207 − 0.360i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 84 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.713 + 0.700i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.36282 - 1.37565i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.36282 - 1.37565i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-51.3 + 62.6i)T \) |
| 7 | \( 1 + (1.25e3 - 2.04e3i)T \) |
| good | 5 | \( 1 + (-918. + 530. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (-2.20e4 - 1.27e4i)T + (1.07e8 + 1.85e8i)T^{2} \) |
| 13 | \( 1 - 4.82e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + (-1.51e4 - 8.72e3i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (-6.76e4 - 1.17e5i)T + (-8.49e9 + 1.47e10i)T^{2} \) |
| 23 | \( 1 + (4.28e5 - 2.47e5i)T + (3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + 4.48e5iT - 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.92e5 + 3.32e5i)T + (-4.26e11 - 7.38e11i)T^{2} \) |
| 37 | \( 1 + (4.05e4 + 7.02e4i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 + 9.81e5iT - 7.98e12T^{2} \) |
| 43 | \( 1 - 1.23e6T + 1.16e13T^{2} \) |
| 47 | \( 1 + (2.72e6 - 1.57e6i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-6.10e5 - 3.52e5i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (1.15e7 + 6.67e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (-5.42e5 - 9.39e5i)T + (-9.58e13 + 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.26e7 + 2.19e7i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 3.64e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + (4.68e6 - 8.10e6i)T + (-4.03e14 - 6.98e14i)T^{2} \) |
| 79 | \( 1 + (-8.25e5 - 1.42e6i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 3.97e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (2.86e7 - 1.65e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 + 1.05e8T + 7.83e15T^{2} \) |
| 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
−12.59712979028593374491707721415, −11.88114407786603916971791524944, −9.696314689184258644119015952283, −9.271792696694358257298306862386, −8.201377158123173869305757557867, −6.34500774522334303943461605990, −5.86678531337013412022362032377, −3.74171056840942327924658131494, −1.94342752231198759826535644719, −1.31381404713263864337341740331,
1.34185978436340509538241738382, 2.99559594392777299468125818306, 3.95688585684239748108077323589, 5.91559733145455724550663959162, 6.73865861340088806781463452960, 8.616861302139691362342123405709, 9.543482006796155897954645200693, 10.42513349949007508027390977556, 11.23492574330973831383223316841, 13.35622155709671460148389155819
