| L(s) = 1 | + 46.7i·3-s − 909.·5-s + 3.38e3i·7-s − 2.18e3·9-s + 2.60e3i·11-s + 1.72e4·13-s − 4.25e4i·15-s + 192.·17-s − 4.35e4i·19-s − 1.58e5·21-s + 1.70e5i·23-s + 4.37e5·25-s − 1.02e5i·27-s + 1.07e6·29-s − 2.78e5i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s − 1.45·5-s + 1.41i·7-s − 0.333·9-s + 0.178i·11-s + 0.605·13-s − 0.840i·15-s + 0.00230·17-s − 0.334i·19-s − 0.814·21-s + 0.608i·23-s + 1.11·25-s − 0.192i·27-s + 1.51·29-s − 0.301i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(1.579497961\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.579497961\) |
| \(L(5)\) |
|
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 | \( 1 + 909.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 3.38e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 2.60e3iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 1.72e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 192.T + 6.97e9T^{2} \) |
| 19 | \( 1 + 4.35e4iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.70e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 - 1.07e6T + 5.00e11T^{2} \) |
| 31 | \( 1 + 2.78e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.51e6T + 3.51e12T^{2} \) |
| 41 | \( 1 - 2.03e4T + 7.98e12T^{2} \) |
| 43 | \( 1 + 3.76e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 6.48e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 7.46e5T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.79e7iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 1.96e7T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.79e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 + 1.59e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 2.27e6T + 8.06e14T^{2} \) |
| 79 | \( 1 + 5.12e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 - 4.37e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 - 7.31e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 3.05e7T + 7.83e15T^{2} \) |
| 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
−10.23791502492238003875681367980, −9.087876434892039751985892874566, −8.487703306866829562516018092311, −7.64065671339079626470631695626, −6.37324493557392666527181745138, −5.30972454416494500401946515644, −4.32970008789167628863610474966, −3.40250909660506864257180816895, −2.38600640795835431797155900548, −0.68132702569195047780680496149,
0.55411460455868980678785407131, 1.12318404192861707024738556142, 2.90764573580037054018074622497, 3.92045282462587202981724820460, 4.59616324985851282990260223016, 6.26386392528575935084881215771, 7.07525893095953520430638688736, 7.932444121542990749418623185145, 8.384310354668405651185515895125, 9.873237168601809339216863046385
