| L(s) = 1 | + (−2.16 + 3.75i)2-s + (11.8 + 10.1i)3-s + (6.59 + 11.4i)4-s + (29.6 + 51.4i)5-s + (−63.8 + 22.2i)6-s + (−30.4 + 52.6i)7-s − 195.·8-s + (35.6 + 240. i)9-s − 257.·10-s + (60.5 − 104. i)11-s + (−38.4 + 202. i)12-s + (11.6 + 20.2i)13-s + (−131. − 228. i)14-s + (−173. + 909. i)15-s + (213. − 370. i)16-s + 520.·17-s + ⋯ |
| L(s) = 1 | + (−0.383 + 0.663i)2-s + (0.757 + 0.653i)3-s + (0.206 + 0.357i)4-s + (0.531 + 0.920i)5-s + (−0.723 + 0.252i)6-s + (−0.234 + 0.406i)7-s − 1.08·8-s + (0.146 + 0.989i)9-s − 0.814·10-s + (0.150 − 0.261i)11-s + (−0.0771 + 0.405i)12-s + (0.0191 + 0.0331i)13-s + (−0.179 − 0.311i)14-s + (−0.198 + 1.04i)15-s + (0.208 − 0.361i)16-s + 0.436·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0272i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 99 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0272i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0278736 + 2.04792i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0278736 + 2.04792i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-11.8 - 10.1i)T \) |
| 11 | \( 1 + (-60.5 + 104. i)T \) |
| good | 2 | \( 1 + (2.16 - 3.75i)T + (-16 - 27.7i)T^{2} \) |
| 5 | \( 1 + (-29.6 - 51.4i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (30.4 - 52.6i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 13 | \( 1 + (-11.6 - 20.2i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 520.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 2.15e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + (825. + 1.42e3i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (942. - 1.63e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (796. + 1.37e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 1.68e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (6.32e3 + 1.09e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (-7.07e3 + 1.22e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (1.28e4 - 2.22e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 + 1.91e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.62e4 - 2.82e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.97e4 + 3.42e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-2.65e4 - 4.60e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 - 3.78e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.37e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (-1.90e3 + 3.29e3i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (-7.87e3 + 1.36e4i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 + 7.80e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (-6.07e3 + 1.05e4i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−13.92479178572511827189308951021, −12.45189217092932929581487814952, −11.13247844241736789313080679113, −9.932557441270753998590053054573, −9.061086359143619040385992516541, −7.945191990551230442921531700582, −6.87174012841295632533120093398, −5.60256054257474448724576503059, −3.49826466746912040238505913223, −2.52089106435546975005149816742,
0.848174059148595226153950052473, 1.79236527788585736357453379601, 3.34662054576670836801282040960, 5.41975005216408973386600153298, 6.79822821816788654498401532537, 8.172792281427691729961406153568, 9.456672089140196505412152094983, 9.825402489415426227874666430398, 11.48177892059238104676572883556, 12.42439475217699630834838718699
