| L(s) = 1 | + (25.4 − 14.6i)2-s + (59.8 − 34.5i)3-s + (302. − 524. i)4-s + (537. + 931. i)5-s + (1.01e3 − 1.75e3i)6-s − 1.76e3·7-s − 1.02e4i·8-s + (−893. + 1.54e3i)9-s + (2.73e4 + 1.57e4i)10-s − 2.37e4·11-s − 4.18e4i·12-s + (−6.87e3 − 3.96e3i)13-s + (−4.49e4 + 2.59e4i)14-s + (6.43e4 + 3.71e4i)15-s + (−7.30e4 − 1.26e5i)16-s + (−8.33e3 − 1.44e4i)17-s + ⋯ |
| L(s) = 1 | + (1.58 − 0.917i)2-s + (0.738 − 0.426i)3-s + (1.18 − 2.04i)4-s + (0.860 + 1.49i)5-s + (0.782 − 1.35i)6-s − 0.736·7-s − 2.50i·8-s + (−0.136 + 0.235i)9-s + (2.73 + 1.57i)10-s − 1.62·11-s − 2.01i·12-s + (−0.240 − 0.138i)13-s + (−1.17 + 0.675i)14-s + (1.27 + 0.734i)15-s + (−1.11 − 1.93i)16-s + (−0.0998 − 0.172i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(9-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+4) \, L(s)\cr =\mathstrut & (0.378 + 0.925i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{9}{2})\) |
\(\approx\) |
\(3.81455 - 2.56258i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.81455 - 2.56258i\) |
| \(L(5)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-1.15e5 + 6.08e4i)T \) |
| good | 2 | \( 1 + (-25.4 + 14.6i)T + (128 - 221. i)T^{2} \) |
| 3 | \( 1 + (-59.8 + 34.5i)T + (3.28e3 - 5.68e3i)T^{2} \) |
| 5 | \( 1 + (-537. - 931. i)T + (-1.95e5 + 3.38e5i)T^{2} \) |
| 7 | \( 1 + 1.76e3T + 5.76e6T^{2} \) |
| 11 | \( 1 + 2.37e4T + 2.14e8T^{2} \) |
| 13 | \( 1 + (6.87e3 + 3.96e3i)T + (4.07e8 + 7.06e8i)T^{2} \) |
| 17 | \( 1 + (8.33e3 + 1.44e4i)T + (-3.48e9 + 6.04e9i)T^{2} \) |
| 23 | \( 1 + (-1.27e5 + 2.20e5i)T + (-3.91e10 - 6.78e10i)T^{2} \) |
| 29 | \( 1 + (-6.59e5 - 3.80e5i)T + (2.50e11 + 4.33e11i)T^{2} \) |
| 31 | \( 1 + 6.40e5iT - 8.52e11T^{2} \) |
| 37 | \( 1 + 1.53e5iT - 3.51e12T^{2} \) |
| 41 | \( 1 + (-3.20e6 + 1.85e6i)T + (3.99e12 - 6.91e12i)T^{2} \) |
| 43 | \( 1 + (5.55e5 + 9.62e5i)T + (-5.84e12 + 1.01e13i)T^{2} \) |
| 47 | \( 1 + (-2.53e6 + 4.39e6i)T + (-1.19e13 - 2.06e13i)T^{2} \) |
| 53 | \( 1 + (2.07e6 + 1.19e6i)T + (3.11e13 + 5.39e13i)T^{2} \) |
| 59 | \( 1 + (4.44e6 - 2.56e6i)T + (7.34e13 - 1.27e14i)T^{2} \) |
| 61 | \( 1 + (6.32e6 - 1.09e7i)T + (-9.58e13 - 1.66e14i)T^{2} \) |
| 67 | \( 1 + (-1.63e7 - 9.46e6i)T + (2.03e14 + 3.51e14i)T^{2} \) |
| 71 | \( 1 + (1.42e7 - 8.23e6i)T + (3.22e14 - 5.59e14i)T^{2} \) |
| 73 | \( 1 + (-1.77e7 - 3.06e7i)T + (-4.03e14 + 6.98e14i)T^{2} \) |
| 79 | \( 1 + (2.60e7 - 1.50e7i)T + (7.58e14 - 1.31e15i)T^{2} \) |
| 83 | \( 1 + 7.99e7T + 2.25e15T^{2} \) |
| 89 | \( 1 + (-1.23e7 - 7.13e6i)T + (1.96e15 + 3.40e15i)T^{2} \) |
| 97 | \( 1 + (6.52e7 - 3.76e7i)T + (3.91e15 - 6.78e15i)T^{2} \) |
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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
−15.61276630284856547817068974245, −14.42958900874610126539620074458, −13.64623579740374108979465648545, −12.87390715105418275691766020784, −10.96706283683947667005180121409, −10.07369564510263539407421425032, −7.04066725718540622564589556593, −5.50761047074302489641785497954, −2.90985135145467337704111904759, −2.51121664048255035824468395960,
2.96690770490033839964116726732, 4.76692056566536055551312910671, 5.90684039529227011321804520306, 8.027910821798047323155548337091, 9.557949023676823048776031319839, 12.38657425677506502007112694644, 13.21232738685189982890666124429, 14.06751860514511701971426784579, 15.61208718753955018885225505926, 16.18968331928941885854136135368
