| L(s) = 1 | + 33.7i·3-s − 51.7·5-s + 313.·7-s − 409.·9-s − 460.·11-s − 177. i·13-s − 1.74e3i·15-s + 6.12e3·17-s + (−5.40e3 + 4.22e3i)19-s + 1.05e4i·21-s − 1.09e4·23-s − 1.29e4·25-s + 1.07e4i·27-s − 1.11e4i·29-s − 1.59e4i·31-s + ⋯ |
| L(s) = 1 | + 1.24i·3-s − 0.414·5-s + 0.913·7-s − 0.561·9-s − 0.346·11-s − 0.0806i·13-s − 0.517i·15-s + 1.24·17-s + (−0.788 + 0.615i)19-s + 1.14i·21-s − 0.896·23-s − 0.828·25-s + 0.547i·27-s − 0.456i·29-s − 0.535i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 304 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.788 + 0.615i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.6587891371\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6587891371\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 19 | \( 1 + (5.40e3 - 4.22e3i)T \) |
| good | 3 | \( 1 - 33.7iT - 729T^{2} \) |
| 5 | \( 1 + 51.7T + 1.56e4T^{2} \) |
| 7 | \( 1 - 313.T + 1.17e5T^{2} \) |
| 11 | \( 1 + 460.T + 1.77e6T^{2} \) |
| 13 | \( 1 + 177. iT - 4.82e6T^{2} \) |
| 17 | \( 1 - 6.12e3T + 2.41e7T^{2} \) |
| 23 | \( 1 + 1.09e4T + 1.48e8T^{2} \) |
| 29 | \( 1 + 1.11e4iT - 5.94e8T^{2} \) |
| 31 | \( 1 + 1.59e4iT - 8.87e8T^{2} \) |
| 37 | \( 1 - 7.65e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 - 1.25e5iT - 4.75e9T^{2} \) |
| 43 | \( 1 - 2.06e4T + 6.32e9T^{2} \) |
| 47 | \( 1 + 1.50e5T + 1.07e10T^{2} \) |
| 53 | \( 1 + 1.13e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + 2.31e5iT - 4.21e10T^{2} \) |
| 61 | \( 1 + 3.14e5T + 5.15e10T^{2} \) |
| 67 | \( 1 - 3.79e5iT - 9.04e10T^{2} \) |
| 71 | \( 1 + 2.18e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 - 3.79e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + 3.31e5iT - 2.43e11T^{2} \) |
| 83 | \( 1 + 1.91e5T + 3.26e11T^{2} \) |
| 89 | \( 1 - 2.68e5iT - 4.96e11T^{2} \) |
| 97 | \( 1 + 1.70e6iT - 8.32e11T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.21281300873319405126630492814, −10.13679651936190644490789233764, −9.727812500035179667779822422159, −8.229533123580166292707896111139, −7.85951603833528659139793719364, −6.14393055891050909200663831125, −5.01961426788968719241994187736, −4.25510105785369495674539901723, −3.26447442311182625615999790746, −1.62077733403582894191215249838,
0.15500536695960822963193816904, 1.38027856816128110699556522864, 2.31679051388675831168038521499, 3.89107811177852144785338359408, 5.20361446987807416823591088632, 6.30253279257863086436907768649, 7.51026583966430097626819423295, 7.83864868940825847996662180563, 8.886044585526940309999789424489, 10.29224087198127252100992663335
