| L(s) = 1 | + (451. − 260. i)2-s + (−6.61e3 − 1.85e4i)3-s + (4.81e3 − 8.34e3i)4-s + (2.21e6 + 1.27e6i)5-s + (−7.81e6 − 6.64e6i)6-s + (2.07e7 + 3.59e7i)7-s + 1.31e8i·8-s + (−2.99e8 + 2.45e8i)9-s + 1.33e9·10-s + (3.20e9 − 1.84e9i)11-s + (−1.86e8 − 3.40e7i)12-s + (−3.00e9 + 5.20e9i)13-s + (1.87e10 + 1.08e10i)14-s + (9.03e9 − 4.94e10i)15-s + (3.55e10 + 6.16e10i)16-s − 1.02e11i·17-s + ⋯ |
| L(s) = 1 | + (0.881 − 0.509i)2-s + (−0.336 − 0.941i)3-s + (0.0183 − 0.0318i)4-s + (1.13 + 0.653i)5-s + (−0.775 − 0.659i)6-s + (0.514 + 0.890i)7-s + 0.980i·8-s + (−0.774 + 0.633i)9-s + 1.33·10-s + (1.35 − 0.784i)11-s + (−0.0361 − 0.00660i)12-s + (−0.283 + 0.491i)13-s + (0.906 + 0.523i)14-s + (0.235 − 1.28i)15-s + (0.517 + 0.896i)16-s − 0.867i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(19-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+9) \, L(s)\cr =\mathstrut & (0.982 + 0.186i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{19}{2})\) |
\(\approx\) |
\(3.28062 - 0.307830i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.28062 - 0.307830i\) |
| \(L(10)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (6.61e3 + 1.85e4i)T \) |
| good | 2 | \( 1 + (-451. + 260. i)T + (1.31e5 - 2.27e5i)T^{2} \) |
| 5 | \( 1 + (-2.21e6 - 1.27e6i)T + (1.90e12 + 3.30e12i)T^{2} \) |
| 7 | \( 1 + (-2.07e7 - 3.59e7i)T + (-8.14e14 + 1.41e15i)T^{2} \) |
| 11 | \( 1 + (-3.20e9 + 1.84e9i)T + (2.77e18 - 4.81e18i)T^{2} \) |
| 13 | \( 1 + (3.00e9 - 5.20e9i)T + (-5.62e19 - 9.73e19i)T^{2} \) |
| 17 | \( 1 + 1.02e11iT - 1.40e22T^{2} \) |
| 19 | \( 1 + 1.19e11T + 1.04e23T^{2} \) |
| 23 | \( 1 + (-1.78e12 - 1.02e12i)T + (1.62e24 + 2.80e24i)T^{2} \) |
| 29 | \( 1 + (-2.24e13 + 1.29e13i)T + (1.05e26 - 1.82e26i)T^{2} \) |
| 31 | \( 1 + (2.21e13 - 3.84e13i)T + (-3.49e26 - 6.05e26i)T^{2} \) |
| 37 | \( 1 + 9.79e13T + 1.68e28T^{2} \) |
| 41 | \( 1 + (-1.24e14 - 7.21e13i)T + (5.35e28 + 9.28e28i)T^{2} \) |
| 43 | \( 1 + (1.20e14 + 2.09e14i)T + (-1.26e29 + 2.18e29i)T^{2} \) |
| 47 | \( 1 + (2.53e14 - 1.46e14i)T + (6.26e29 - 1.08e30i)T^{2} \) |
| 53 | \( 1 + 1.01e15iT - 1.08e31T^{2} \) |
| 59 | \( 1 + (1.18e16 + 6.83e15i)T + (3.75e31 + 6.49e31i)T^{2} \) |
| 61 | \( 1 + (-6.12e15 - 1.06e16i)T + (-6.83e31 + 1.18e32i)T^{2} \) |
| 67 | \( 1 + (-9.37e15 + 1.62e16i)T + (-3.70e32 - 6.41e32i)T^{2} \) |
| 71 | \( 1 - 1.68e16iT - 2.10e33T^{2} \) |
| 73 | \( 1 - 6.33e16T + 3.46e33T^{2} \) |
| 79 | \( 1 + (1.71e16 + 2.96e16i)T + (-7.18e33 + 1.24e34i)T^{2} \) |
| 83 | \( 1 + (5.38e16 - 3.11e16i)T + (1.74e34 - 3.02e34i)T^{2} \) |
| 89 | \( 1 + 1.85e17iT - 1.22e35T^{2} \) |
| 97 | \( 1 + (2.02e17 + 3.51e17i)T + (-2.88e35 + 5.00e35i)T^{2} \) |
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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
−17.12348627908636404760794734075, −14.35003325107838434763166389428, −13.77116253860076782882118096959, −12.19400669241907470474034104272, −11.25403965045902914338608559201, −8.812334870373178814866601284627, −6.55807371094063564925511685425, −5.26351337880887771252711719025, −2.83551368584149802227849107414, −1.66770168414667078040908227667,
1.12891086851492638395099481009, 4.09865578012921240561101934062, 5.07835964543302547589912571946, 6.46155392104824522351434579886, 9.263401789549096339951337829390, 10.44471419464478945400881396678, 12.63707720576092498347876837350, 14.13430437424525458135331945451, 14.99723268059923532435285853696, 16.79525157116941557737721411058
