| L(s) = 1 | − 46.7i·3-s − 1.07e3·5-s + 3.98e3i·7-s − 2.18e3·9-s − 2.01e4i·11-s + 3.61e4·13-s + 5.04e4i·15-s + 5.99e4·17-s + 1.26e5i·19-s + 1.86e5·21-s + 1.97e5i·23-s + 7.73e5·25-s + 1.02e5i·27-s − 4.85e5·29-s − 6.71e5i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 1.72·5-s + 1.65i·7-s − 0.333·9-s − 1.37i·11-s + 1.26·13-s + 0.996i·15-s + 0.717·17-s + 0.969i·19-s + 0.957·21-s + 0.704i·23-s + 1.98·25-s + 0.192i·27-s − 0.686·29-s − 0.727i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 192 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.03603980658\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.03603980658\) |
| \(L(5)\) |
|
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 | \( 1 + 1.07e3T + 3.90e5T^{2} \) |
| 7 | \( 1 - 3.98e3iT - 5.76e6T^{2} \) |
| 11 | \( 1 + 2.01e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 - 3.61e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 5.99e4T + 6.97e9T^{2} \) |
| 19 | \( 1 - 1.26e5iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 1.97e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.85e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 6.71e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.40e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.55e6T + 7.98e12T^{2} \) |
| 43 | \( 1 - 7.36e5iT - 1.16e13T^{2} \) |
| 47 | \( 1 - 8.83e5iT - 2.38e13T^{2} \) |
| 53 | \( 1 + 1.44e7T + 6.22e13T^{2} \) |
| 59 | \( 1 - 1.08e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 - 3.45e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 1.43e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 3.97e6iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 3.92e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 5.94e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 5.42e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 2.43e7T + 3.93e15T^{2} \) |
| 97 | \( 1 + 3.83e7T + 7.83e15T^{2} \) |
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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
−11.54112387777514024252932325901, −11.13547776115167225376910499260, −9.228823612563671105341392740821, −8.163557256609689424982507985628, −7.990879493953645575524962988669, −6.28352962005440026491045428159, −5.52440301519098962348664619502, −3.76932919742649899426585999089, −2.99613581779731923310921133084, −1.28634864296724289621864881993,
0.01055509507755230018429003437, 1.08678238847418017853074849109, 3.30341068305947984790436780843, 4.11775054615874704078256055293, 4.71727955311635159294241808464, 6.75809799038239027003319411935, 7.52465778320538417055888679867, 8.362548323551151671658998025080, 9.727589852624444543515026639710, 10.76375217799787581391759684432
