| L(s) = 1 | + (−12.6 − 21.9i)2-s + (7.68 + 4.43i)3-s + (−193. + 334. i)4-s + (538. − 310. i)5-s − 224. i·6-s + 3.29e3·8-s + (−3.24e3 − 5.61e3i)9-s + (−1.36e4 − 7.87e3i)10-s + (−7.54e3 + 1.30e4i)11-s + (−2.96e3 + 1.71e3i)12-s + 4.51e4i·13-s + 5.51e3·15-s + (7.65e3 + 1.32e4i)16-s + (−3.59e4 − 2.07e4i)17-s + (−8.21e4 + 1.42e5i)18-s + (−1.02e5 + 5.90e4i)19-s + ⋯ |
| L(s) = 1 | + (−0.791 − 1.37i)2-s + (0.0948 + 0.0547i)3-s + (−0.754 + 1.30i)4-s + (0.861 − 0.497i)5-s − 0.173i·6-s + 0.804·8-s + (−0.494 − 0.855i)9-s + (−1.36 − 0.787i)10-s + (−0.515 + 0.892i)11-s + (−0.143 + 0.0825i)12-s + 1.57i·13-s + 0.108·15-s + (0.116 + 0.202i)16-s + (−0.430 − 0.248i)17-s + (−0.782 + 1.35i)18-s + (−0.784 + 0.453i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 49 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.895 - 0.444i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(0.583100 + 0.136597i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.583100 + 0.136597i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| good | 2 | \( 1 + (12.6 + 21.9i)T + (-128 + 221. i)T^{2} \) |
| 3 | \( 1 + (-7.68 - 4.43i)T + (3.28e3 + 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-538. + 310. i)T + (1.95e5 - 3.38e5i)T^{2} \) |
| 11 | \( 1 + (7.54e3 - 1.30e4i)T + (-1.07e8 - 1.85e8i)T^{2} \) |
| 13 | \( 1 - 4.51e4iT - 8.15e8T^{2} \) |
| 17 | \( 1 + (3.59e4 + 2.07e4i)T + (3.48e9 + 6.04e9i)T^{2} \) |
| 19 | \( 1 + (1.02e5 - 5.90e4i)T + (8.49e9 - 1.47e10i)T^{2} \) |
| 23 | \( 1 + (1.08e5 + 1.87e5i)T + (-3.91e10 + 6.78e10i)T^{2} \) |
| 29 | \( 1 + 3.02e4T + 5.00e11T^{2} \) |
| 31 | \( 1 + (-1.11e6 - 6.42e5i)T + (4.26e11 + 7.38e11i)T^{2} \) |
| 37 | \( 1 + (-1.31e6 - 2.27e6i)T + (-1.75e12 + 3.04e12i)T^{2} \) |
| 41 | \( 1 - 1.05e6iT - 7.98e12T^{2} \) |
| 43 | \( 1 + 6.68e5T + 1.16e13T^{2} \) |
| 47 | \( 1 + (8.46e5 - 4.88e5i)T + (1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (-2.36e5 + 4.08e5i)T + (-3.11e13 - 5.39e13i)T^{2} \) |
| 59 | \( 1 + (-1.62e7 - 9.35e6i)T + (7.34e13 + 1.27e14i)T^{2} \) |
| 61 | \( 1 + (2.08e7 - 1.20e7i)T + (9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-5.58e6 + 9.67e6i)T + (-2.03e14 - 3.51e14i)T^{2} \) |
| 71 | \( 1 + 7.52e6T + 6.45e14T^{2} \) |
| 73 | \( 1 + (8.54e6 + 4.93e6i)T + (4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (7.33e6 + 1.27e7i)T + (-7.58e14 + 1.31e15i)T^{2} \) |
| 83 | \( 1 - 3.82e6iT - 2.25e15T^{2} \) |
| 89 | \( 1 + (3.43e7 - 1.98e7i)T + (1.96e15 - 3.40e15i)T^{2} \) |
| 97 | \( 1 - 7.73e7iT - 7.83e15T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.59197170917638861264441350888, −12.43070465774669782709310951679, −11.61319401396921189407224169507, −10.18829867195448746380859699648, −9.403210536079681162574900632755, −8.533494171120728304267159046294, −6.40509546430772968164784532236, −4.37331779035765038258940371948, −2.53424289482346978841317765663, −1.43326399634345469778062418909,
0.27658203061764541978891772930, 2.61833325889103598661070692826, 5.44301400275086184221719737287, 6.19932378971716312131300104483, 7.74673502056553455660168055040, 8.527740590624485670296133733716, 9.981174150785258518613686405930, 10.92890054600712085521635640765, 13.16840603796112593869025177363, 14.04298179088162651788322039329
