| L(s) = 1 | − 9i·3-s − 40.9·5-s − 89.5·7-s − 81·9-s + (−199. − 348. i)11-s − 61.8i·13-s + 368. i·15-s − 1.93e3i·17-s + 411.·19-s + 805. i·21-s − 2.58e3i·23-s − 1.44e3·25-s + 729i·27-s − 6.74e3i·29-s + 1.05e3i·31-s + ⋯ |
| L(s) = 1 | − 0.577i·3-s − 0.733·5-s − 0.690·7-s − 0.333·9-s + (−0.497 − 0.867i)11-s − 0.101i·13-s + 0.423i·15-s − 1.62i·17-s + 0.261·19-s + 0.398i·21-s − 1.01i·23-s − 0.462·25-s + 0.192i·27-s − 1.49i·29-s + 0.197i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.00308 - 0.999i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 528 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.00308 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.08653936535\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.08653936535\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + 9iT \) |
| 11 | \( 1 + (199. + 348. i)T \) |
| good | 5 | \( 1 + 40.9T + 3.12e3T^{2} \) |
| 7 | \( 1 + 89.5T + 1.68e4T^{2} \) |
| 13 | \( 1 + 61.8iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.93e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 411.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 2.58e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 6.74e3iT - 2.05e7T^{2} \) |
| 31 | \( 1 - 1.05e3iT - 2.86e7T^{2} \) |
| 37 | \( 1 - 849.T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.30e4iT - 1.15e8T^{2} \) |
| 43 | \( 1 + 4.12e3T + 1.47e8T^{2} \) |
| 47 | \( 1 + 8.47e3iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 9.96e3T + 4.18e8T^{2} \) |
| 59 | \( 1 - 3.09e4iT - 7.14e8T^{2} \) |
| 61 | \( 1 + 2.76e4iT - 8.44e8T^{2} \) |
| 67 | \( 1 - 5.54e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 552. iT - 1.80e9T^{2} \) |
| 73 | \( 1 + 2.32e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 3.07e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 5.04e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.21e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 9.95e4T + 8.58e9T^{2} \) |
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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
−9.363129984080405042483508962869, −8.311018334486498300913265459219, −7.64453441477374536936810579662, −6.70743875023135564723564304401, −5.81075974033221181433712212425, −4.62951919202108182252334858472, −3.33384695922692384520761257001, −2.53280584166605341491308395680, −0.73973971773221508632159137733, −0.02773386369907931773583963472,
1.75418524803222625937107638598, 3.26429472112215527778276194972, 3.96686036322090554317099536904, 5.05093487928364208654379824575, 6.10963056606409860941152152383, 7.22709964705555603521254752139, 8.038634529335698233933737389047, 9.046937304607698859441559586764, 9.910565858105276294090188486650, 10.60980527397395940273847806237
