| L(s) = 1 | − 10.2i·3-s + (17.9 − 17.4i)5-s − 84.8·7-s − 23.1·9-s − 71.3·11-s − 109.·13-s + (−177. − 182. i)15-s + 151. i·17-s + 368.·19-s + 865. i·21-s − 358.·23-s + (16.8 − 624. i)25-s − 590. i·27-s + 387. i·29-s + 1.68e3i·31-s + ⋯ |
| L(s) = 1 | − 1.13i·3-s + (0.716 − 0.697i)5-s − 1.73·7-s − 0.285·9-s − 0.589·11-s − 0.646·13-s + (−0.790 − 0.812i)15-s + 0.523i·17-s + 1.02·19-s + 1.96i·21-s − 0.676·23-s + (0.0269 − 0.999i)25-s − 0.810i·27-s + 0.460i·29-s + 1.75i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 160 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (-0.780 - 0.624i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.167450 + 0.477457i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.167450 + 0.477457i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + (-17.9 + 17.4i)T \) |
| good | 3 | \( 1 + 10.2iT - 81T^{2} \) |
| 7 | \( 1 + 84.8T + 2.40e3T^{2} \) |
| 11 | \( 1 + 71.3T + 1.46e4T^{2} \) |
| 13 | \( 1 + 109.T + 2.85e4T^{2} \) |
| 17 | \( 1 - 151. iT - 8.35e4T^{2} \) |
| 19 | \( 1 - 368.T + 1.30e5T^{2} \) |
| 23 | \( 1 + 358.T + 2.79e5T^{2} \) |
| 29 | \( 1 - 387. iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.68e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.15e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 2.54e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + 1.35e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 901.T + 4.87e6T^{2} \) |
| 53 | \( 1 + 3.40e3T + 7.89e6T^{2} \) |
| 59 | \( 1 - 1.45e3T + 1.21e7T^{2} \) |
| 61 | \( 1 + 5.32e3iT - 1.38e7T^{2} \) |
| 67 | \( 1 - 657. iT - 2.01e7T^{2} \) |
| 71 | \( 1 + 6.74e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 - 4.13e3iT - 2.83e7T^{2} \) |
| 79 | \( 1 + 2.30e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 - 2.14e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 5.11e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 9.16e3iT - 8.85e7T^{2} \) |
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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
−12.28392788282711924923646457859, −10.28678013216100631515210290454, −9.633611130069341825072229875556, −8.436278327604428446927356029757, −7.13385187525417763573819307870, −6.35784458249013753850175184130, −5.20719071277198823039279497246, −3.18681882498454037845828943716, −1.72602965161872832352622484820, −0.18151695912672010656135035554,
2.66883107020314396175123663635, 3.64215788105555989789201969431, 5.21202061316519386308961140350, 6.31221508928891963700396825934, 7.42355547421159676252372240396, 9.326848207539226517462108199509, 9.863249492008954125013468974274, 10.32094542274548057133967279960, 11.67087755936826612467748901875, 12.99587414395181417205107410465
