| L(s) = 1 | + 46.7i·3-s + 679.·5-s − 2.83e3i·7-s − 2.18e3·9-s + 1.97e4i·11-s + 9.31e3·13-s + 3.17e4i·15-s − 1.18e4·17-s − 1.86e5i·19-s + 1.32e5·21-s + 1.61e5i·23-s + 7.06e4·25-s − 1.02e5i·27-s + 3.64e5·29-s + 1.63e6i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 1.08·5-s − 1.17i·7-s − 0.333·9-s + 1.34i·11-s + 0.325·13-s + 0.627i·15-s − 0.142·17-s − 1.43i·19-s + 0.680·21-s + 0.577i·23-s + 0.180·25-s − 0.192i·27-s + 0.515·29-s + 1.76i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(2.327805892\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.327805892\) |
| \(L(5)\) |
|
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 | \( 1 - 679.T + 3.90e5T^{2} \) |
| 7 | \( 1 + 2.83e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.97e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 9.31e3T + 8.15e8T^{2} \) |
| 17 | \( 1 + 1.18e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 1.86e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.61e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 3.64e5T + 5.00e11T^{2} \) |
| 31 | \( 1 - 1.63e6iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.25e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 4.90e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 6.24e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 1.78e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 6.12e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.94e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 2.08e7T + 1.91e14T^{2} \) |
| 67 | \( 1 - 4.51e6iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.25e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.10e7T + 8.06e14T^{2} \) |
| 79 | \( 1 + 2.98e6iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 3.64e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 8.65e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 6.39e7T + 7.83e15T^{2} \) |
| 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
−10.13344324935297440438521846569, −9.561744850791684893106724707178, −8.567808874086612468655235306151, −7.18409545692922063865151079530, −6.60328856923477440787475242080, −5.20113278106649897312834690328, −4.55089062651903015326840006608, −3.36144892407185555391379706558, −2.08552925884896760171960299294, −1.04430126311311596597564667190,
0.45282200319125751174690184318, 1.73621373193572461149767729113, 2.45093602896795460219422898286, 3.63845481015913823103887361957, 5.48373937004087681388247729259, 5.82716700139633437296841664741, 6.66480305281691743393664676360, 8.204297547294263482358042980256, 8.668329608527573190370687575324, 9.691719475189898027832767817566
