| L(s) = 1 | − 141.·3-s − 2.55e3·7-s + 1.35e4·9-s − 1.91e4i·11-s + 2.77e4i·13-s + 5.03e4i·17-s + 1.08e5i·19-s + 3.62e5·21-s − 1.76e5·23-s − 9.99e5·27-s − 5.49e4·29-s − 1.17e6i·31-s + 2.72e6i·33-s + 7.93e5i·37-s − 3.93e6i·39-s + ⋯ |
| L(s) = 1 | − 1.75·3-s − 1.06·7-s + 2.07·9-s − 1.30i·11-s + 0.970i·13-s + 0.603i·17-s + 0.833i·19-s + 1.86·21-s − 0.630·23-s − 1.88·27-s − 0.0777·29-s − 1.27i·31-s + 2.29i·33-s + 0.423i·37-s − 1.70i·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.447 - 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.3153613718\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3153613718\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 141.T + 6.56e3T^{2} \) |
| 7 | \( 1 + 2.55e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 1.91e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 2.77e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 - 5.03e4iT - 6.97e9T^{2} \) |
| 19 | \( 1 - 1.08e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 + 1.76e5T + 7.83e10T^{2} \) |
| 29 | \( 1 + 5.49e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + 1.17e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 7.93e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + 7.55e4T + 7.98e12T^{2} \) |
| 43 | \( 1 + 4.99e5T + 1.16e13T^{2} \) |
| 47 | \( 1 - 2.86e6T + 2.38e13T^{2} \) |
| 53 | \( 1 + 1.11e7iT - 6.22e13T^{2} \) |
| 59 | \( 1 + 2.18e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 2.38e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 7.49e6T + 4.06e14T^{2} \) |
| 71 | \( 1 - 1.00e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 + 6.51e6iT - 8.06e14T^{2} \) |
| 79 | \( 1 + 4.87e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 7.34e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + 8.67e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 4.66e7iT - 7.83e15T^{2} \) |
| 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
−10.21328150003355463154438608886, −9.495448649156679693526192953002, −8.185405493521239945862048778387, −6.86952924062563360387093448046, −6.13226942048627865111764829523, −5.72421239097499363557047047350, −4.38936192738673117353868498056, −3.45134932454943720710210562919, −1.70127019855436247034101360573, −0.46960785967198620690879698889,
0.17466674928684781887001950402, 1.23008708650207406322403643795, 2.78064903969426571506048786112, 4.24327765136148276801356146872, 5.12383968055934745020149707987, 5.95552904729897319097058077703, 6.84846903417362738737156843140, 7.45528701882820196510723395268, 9.172234247552365790896124585366, 10.08334744595124484361985211624
