| L(s) = 1 | + (−1.97 + 6.06i)2-s + (4.72 − 14.8i)3-s + (−7.05 − 5.12i)4-s + (−59.6 + 19.3i)5-s + (80.8 + 57.9i)6-s + (−97.0 + 133. i)7-s + (−120. + 87.3i)8-s + (−198. − 140. i)9-s − 400. i·10-s + (390. + 92.8i)11-s + (−109. + 80.5i)12-s + (−1.02e3 − 331. i)13-s + (−619. − 852. i)14-s + (6.06 + 978. i)15-s + (−379. − 1.16e3i)16-s + (77.1 + 237. i)17-s + ⋯ |
| L(s) = 1 | + (−0.348 + 1.07i)2-s + (0.303 − 0.952i)3-s + (−0.220 − 0.160i)4-s + (−1.06 + 0.346i)5-s + (0.916 + 0.657i)6-s + (−0.748 + 1.03i)7-s + (−0.663 + 0.482i)8-s + (−0.816 − 0.577i)9-s − 1.26i·10-s + (0.972 + 0.231i)11-s + (−0.219 + 0.161i)12-s + (−1.67 − 0.544i)13-s + (−0.844 − 1.16i)14-s + (0.00696 + 1.12i)15-s + (−0.370 − 1.13i)16-s + (0.0647 + 0.199i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 + 0.0649i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.997 + 0.0649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.0173862 - 0.534962i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0173862 - 0.534962i\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-4.72 + 14.8i)T \) |
| 11 | \( 1 + (-390. - 92.8i)T \) |
| good | 2 | \( 1 + (1.97 - 6.06i)T + (-25.8 - 18.8i)T^{2} \) |
| 5 | \( 1 + (59.6 - 19.3i)T + (2.52e3 - 1.83e3i)T^{2} \) |
| 7 | \( 1 + (97.0 - 133. i)T + (-5.19e3 - 1.59e4i)T^{2} \) |
| 13 | \( 1 + (1.02e3 + 331. i)T + (3.00e5 + 2.18e5i)T^{2} \) |
| 17 | \( 1 + (-77.1 - 237. i)T + (-1.14e6 + 8.34e5i)T^{2} \) |
| 19 | \( 1 + (-1.75e3 - 2.41e3i)T + (-7.65e5 + 2.35e6i)T^{2} \) |
| 23 | \( 1 + 1.76e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + (-2.05e3 - 1.49e3i)T + (6.33e6 + 1.95e7i)T^{2} \) |
| 31 | \( 1 + (-547. + 1.68e3i)T + (-2.31e7 - 1.68e7i)T^{2} \) |
| 37 | \( 1 + (-1.79e3 - 1.30e3i)T + (2.14e7 + 6.59e7i)T^{2} \) |
| 41 | \( 1 + (-69.9 + 50.8i)T + (3.58e7 - 1.10e8i)T^{2} \) |
| 43 | \( 1 - 461. iT - 1.47e8T^{2} \) |
| 47 | \( 1 + (-9.13e3 - 1.25e4i)T + (-7.08e7 + 2.18e8i)T^{2} \) |
| 53 | \( 1 + (1.08e4 + 3.51e3i)T + (3.38e8 + 2.45e8i)T^{2} \) |
| 59 | \( 1 + (8.50e3 - 1.16e4i)T + (-2.20e8 - 6.79e8i)T^{2} \) |
| 61 | \( 1 + (2.50e4 - 8.15e3i)T + (6.83e8 - 4.96e8i)T^{2} \) |
| 67 | \( 1 + 1.67e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + (5.46e4 - 1.77e4i)T + (1.45e9 - 1.06e9i)T^{2} \) |
| 73 | \( 1 + (-5.08e3 + 6.99e3i)T + (-6.40e8 - 1.97e9i)T^{2} \) |
| 79 | \( 1 + (-2.61e3 - 850. i)T + (2.48e9 + 1.80e9i)T^{2} \) |
| 83 | \( 1 + (1.05e4 + 3.23e4i)T + (-3.18e9 + 2.31e9i)T^{2} \) |
| 89 | \( 1 - 3.25e4iT - 5.58e9T^{2} \) |
| 97 | \( 1 + (3.36e4 - 1.03e5i)T + (-6.94e9 - 5.04e9i)T^{2} \) |
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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
−16.24230330843176407907627209486, −15.05262115363544361801121121232, −14.47857075355908348132808763122, −12.17180574632438242928364266732, −12.09278547182627075580182626384, −9.392873643707040994904146591951, −8.034907360962040497009062948747, −7.19409175594737693663542154897, −5.96185510277189593962840496076, −2.94236616227500713317438695269,
0.33405560972954562305508274232, 3.16041334585588539234073607573, 4.39794489856616802874827451129, 7.19707831566045391575761692640, 9.178179594203328810089053057226, 9.924043576320363820281187714546, 11.29570693139417941737763026470, 12.07075882095862481215328503259, 13.82906494758115897834887503142, 15.26664582636421948755190796402
