| L(s) = 1 | + 46.7i·3-s + 441.·5-s + 408. i·7-s − 2.18e3·9-s + 1.39e4i·11-s − 5.05e4·13-s + 2.06e4i·15-s + 2.93e4·17-s − 465. i·19-s − 1.91e4·21-s + 3.49e5i·23-s − 1.95e5·25-s − 1.02e5i·27-s − 4.13e5·29-s − 2.91e5i·31-s + ⋯ |
| L(s) = 1 | + 0.577i·3-s + 0.706·5-s + 0.170i·7-s − 0.333·9-s + 0.955i·11-s − 1.76·13-s + 0.408i·15-s + 0.351·17-s − 0.00357i·19-s − 0.0982·21-s + 1.24i·23-s − 0.500·25-s − 0.192i·27-s − 0.584·29-s − 0.315i·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
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.4970238701\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4970238701\) |
| \(L(5)\) |
|
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 | \( 1 - 441.T + 3.90e5T^{2} \) |
| 7 | \( 1 - 408. iT - 5.76e6T^{2} \) |
| 11 | \( 1 - 1.39e4iT - 2.14e8T^{2} \) |
| 13 | \( 1 + 5.05e4T + 8.15e8T^{2} \) |
| 17 | \( 1 - 2.93e4T + 6.97e9T^{2} \) |
| 19 | \( 1 + 465. iT - 1.69e10T^{2} \) |
| 23 | \( 1 - 3.49e5iT - 7.83e10T^{2} \) |
| 29 | \( 1 + 4.13e5T + 5.00e11T^{2} \) |
| 31 | \( 1 + 2.91e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 - 3.18e6T + 3.51e12T^{2} \) |
| 41 | \( 1 + 3.90e6T + 7.98e12T^{2} \) |
| 43 | \( 1 + 5.13e6iT - 1.16e13T^{2} \) |
| 47 | \( 1 + 3.80e6iT - 2.38e13T^{2} \) |
| 53 | \( 1 - 1.21e6T + 6.22e13T^{2} \) |
| 59 | \( 1 - 5.53e6iT - 1.46e14T^{2} \) |
| 61 | \( 1 + 8.14e6T + 1.91e14T^{2} \) |
| 67 | \( 1 + 2.31e7iT - 4.06e14T^{2} \) |
| 71 | \( 1 - 1.37e7iT - 6.45e14T^{2} \) |
| 73 | \( 1 - 5.32e6T + 8.06e14T^{2} \) |
| 79 | \( 1 - 3.42e7iT - 1.51e15T^{2} \) |
| 83 | \( 1 + 2.71e7iT - 2.25e15T^{2} \) |
| 89 | \( 1 + 3.90e7T + 3.93e15T^{2} \) |
| 97 | \( 1 - 4.04e6T + 7.83e15T^{2} \) |
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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
−9.751907322166871769986048726512, −9.227493567164763972616130720879, −7.80780654382878051274744056608, −7.02986633550956173762510580030, −5.68121389780171030633736706482, −5.04167032788939878226548472897, −3.94952015690818713220191146797, −2.59558246915418369594606325754, −1.78384563643076775576522049020, −0.098545753525344237440429534502,
0.954258095695544562184852359387, 2.17957744631318382508249867252, 2.99492134699669761788870025764, 4.53053666405014176856146841945, 5.60002449260230783543046482240, 6.41689064219087359227052685639, 7.43092451885954484163753332950, 8.263098987156642106655932974385, 9.392196539899860136878186380435, 10.10213005502242375262714500491
