L(s) = 1 | − 46.7i·3-s + 19.7·5-s + 759. i·7-s − 2.18e3·9-s + 1.58e4i·11-s + 4.76e4·13-s − 923. i·15-s − 1.17e5·17-s − 1.96e5i·19-s + 3.55e4·21-s − 1.43e5i·23-s − 3.90e5·25-s + 1.02e5i·27-s − 1.30e6·29-s + 1.08e6i·31-s + ⋯ |
L(s) = 1 | − 0.577i·3-s + 0.0315·5-s + 0.316i·7-s − 0.333·9-s + 1.08i·11-s + 1.66·13-s − 0.0182i·15-s − 1.41·17-s − 1.51i·19-s + 0.182·21-s − 0.513i·23-s − 0.999·25-s + 0.192i·27-s − 1.84·29-s + 1.17i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{9}{2})\) |
\(\approx\) |
\(1.620663719\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.620663719\) |
\(L(5)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + 46.7iT \) |
good | 5 | \( 1 - 19.7T + 3.90e5T^{2} \) |
| 7 | \( 1 - 759. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.58e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 4.76e4T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.17e5T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.96e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.43e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 1.30e6T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.08e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 2.61e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 9.56e5T + 7.98e12T^{2} \) |
| 43 | \( 1 - 4.78e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 2.03e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 4.50e6T + 6.22e13T^{2} \) |
| 59 | \( 1 + 1.40e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.05e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.77e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 3.22e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.34e7T + 8.06e14T^{2} \) |
| 79 | \( 1 - 1.08e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 7.99e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 7.46e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 5.73e7T + 7.83e15T^{2} \) |
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
−9.522322782058451688610268237083, −8.871569706639532940310633528429, −7.88808843628557903688315312514, −6.83350445084927473140618294711, −6.18212034376316096603017477679, −4.93715628302442784058707380290, −3.87343118614178949518479093013, −2.48345841984340777598457097847, −1.65152621094452251911685317975, −0.36983079361367700288604468248,
0.865665956840751443924633038349, 2.14263444761688898095075562751, 3.74662713040964370382923811430, 3.96107969596977962471873602706, 5.72953120594136140104198024676, 6.07160847595184640813014997867, 7.58042949029871846144574790906, 8.502078915941185477468017539733, 9.236404523223688667618836006005, 10.28314778657668033092173904802