| L(s) = 1 | + (−5.09 + 14.7i)3-s − 49.7i·5-s − 155. i·7-s + (−191. − 150. i)9-s − 121·11-s + 866.·13-s + (733. + 253. i)15-s − 225. i·17-s + 557. i·19-s + (2.28e3 + 790. i)21-s + 1.00e3·23-s + 645.·25-s + (3.18e3 − 2.05e3i)27-s + 2.51e3i·29-s + 1.09e3i·31-s + ⋯ |
| L(s) = 1 | + (−0.326 + 0.945i)3-s − 0.890i·5-s − 1.19i·7-s + (−0.786 − 0.617i)9-s − 0.301·11-s + 1.42·13-s + (0.841 + 0.290i)15-s − 0.189i·17-s + 0.353i·19-s + (1.13 + 0.391i)21-s + 0.395·23-s + 0.206·25-s + (0.840 − 0.541i)27-s + 0.554i·29-s + 0.204i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.326 + 0.945i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.331082773\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.331082773\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (5.09 - 14.7i)T \) |
| 11 | \( 1 + 121T \) |
| good | 5 | \( 1 + 49.7iT - 3.12e3T^{2} \) |
| 7 | \( 1 + 155. iT - 1.68e4T^{2} \) |
| 13 | \( 1 - 866.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 225. iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 557. iT - 2.47e6T^{2} \) |
| 23 | \( 1 - 1.00e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 2.51e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.09e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 7.04e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + 2.01e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 2.60e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 2.60e4T + 2.29e8T^{2} \) |
| 53 | \( 1 + 1.24e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 + 2.72e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.90e3T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.51e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 1.13e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 3.16e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 3.42e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 - 7.14e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 2.25e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + 1.58e4T + 8.58e9T^{2} \) |
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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
−9.883740840057020161164178294710, −8.922952033914264542797553720662, −8.269382782703095278627614986571, −7.01635780435641152602831000139, −5.90003504013203775152411556203, −4.95784690312310695815063628509, −4.12359648639340936290873225622, −3.30373911950289234567302457892, −1.29968579440699143031745757398, −0.35426603175636207912535705147,
1.21644987355807615919429339490, 2.42323507571384021088687964275, 3.20950274937666291097735061001, 4.93965780984489735517802774835, 6.18794986700312615379725860953, 6.34034367507478266599184841972, 7.65773753304613794722987769668, 8.394616723314757446813962482440, 9.311807892213092103988974622879, 10.63203169700909314208428059520
