| L(s) = 1 | + (7.46 + 13.6i)3-s − 31.0i·5-s + 49i·7-s + (−131. + 204. i)9-s − 273.·11-s + 431.·13-s + (424. − 231. i)15-s − 648. i·17-s + 2.30e3i·19-s + (−670. + 365. i)21-s − 3.72e3·23-s + 2.16e3·25-s + (−3.77e3 − 274. i)27-s − 7.00e3i·29-s − 1.72e3i·31-s + ⋯ |
| L(s) = 1 | + (0.478 + 0.877i)3-s − 0.554i·5-s + 0.377i·7-s + (−0.541 + 0.840i)9-s − 0.680·11-s + 0.708·13-s + (0.486 − 0.265i)15-s − 0.544i·17-s + 1.46i·19-s + (−0.331 + 0.181i)21-s − 1.46·23-s + 0.692·25-s + (−0.997 − 0.0723i)27-s − 1.54i·29-s − 0.322i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.877 + 0.478i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 336 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.877 + 0.478i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.3339009575\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3339009575\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-7.46 - 13.6i)T \) |
| 7 | \( 1 - 49iT \) |
| good | 5 | \( 1 + 31.0iT - 3.12e3T^{2} \) |
| 11 | \( 1 + 273.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 431.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 648. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 2.30e3iT - 2.47e6T^{2} \) |
| 23 | \( 1 + 3.72e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 7.00e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 + 1.72e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 + 1.22e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.66e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 - 9.18e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.32e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.91e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 1.03e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 7.05e3T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.61e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 7.59e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.11e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 1.06e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + 6.75e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 7.33e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.34e5T + 8.58e9T^{2} \) |
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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
−11.17579600380905602445789804103, −10.09174510281557977523225898419, −9.552141261213874526795997810323, −8.291304919963697055381775300155, −8.032373593668647442421816733543, −6.18281585698424364479775947137, −5.24471731686924473128702197631, −4.23619358941284975266724805510, −3.13814949349597353669218819478, −1.81378786769886988003152499526,
0.07511882536934445617476663968, 1.52033833414146352840602580179, 2.74284159772378680600990647584, 3.74790566353240646886725287543, 5.36146837694563894746336443621, 6.61464835712973729913131630157, 7.18331704532985740648226745286, 8.278711639798149369710830724868, 8.993667284505364523637711841171, 10.38053867858025240347141726369
