| L(s) = 1 | + (−3.83 + 6.64i)2-s + (111. − 193. i)3-s + (226. + 392. i)4-s + (−746. + 1.29e3i)5-s + (857. + 1.48e3i)6-s + 8.35e3·7-s − 7.40e3·8-s + (−1.51e4 − 2.62e4i)9-s + (−5.72e3 − 9.91e3i)10-s + 5.87e4·11-s + 1.01e5·12-s + (3.13e4 + 5.42e4i)13-s + (−3.20e4 + 5.55e4i)14-s + (1.66e5 + 2.88e5i)15-s + (−8.75e4 + 1.51e5i)16-s + (2.80e5 − 4.86e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.169 + 0.293i)2-s + (0.796 − 1.37i)3-s + (0.442 + 0.766i)4-s + (−0.533 + 0.924i)5-s + (0.270 + 0.467i)6-s + 1.31·7-s − 0.639·8-s + (−0.768 − 1.33i)9-s + (−0.181 − 0.313i)10-s + 1.21·11-s + 1.40·12-s + (0.304 + 0.526i)13-s + (−0.223 + 0.386i)14-s + (0.850 + 1.47i)15-s + (−0.334 + 0.578i)16-s + (0.814 − 1.41i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(10-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+9/2) \, L(s)\cr =\mathstrut & (0.982 - 0.184i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(5)\) |
\(\approx\) |
\(2.35560 + 0.218794i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.35560 + 0.218794i\) |
| \(L(\frac{11}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-2.70e5 - 4.99e5i)T \) |
| good | 2 | \( 1 + (3.83 - 6.64i)T + (-256 - 443. i)T^{2} \) |
| 3 | \( 1 + (-111. + 193. i)T + (-9.84e3 - 1.70e4i)T^{2} \) |
| 5 | \( 1 + (746. - 1.29e3i)T + (-9.76e5 - 1.69e6i)T^{2} \) |
| 7 | \( 1 - 8.35e3T + 4.03e7T^{2} \) |
| 11 | \( 1 - 5.87e4T + 2.35e9T^{2} \) |
| 13 | \( 1 + (-3.13e4 - 5.42e4i)T + (-5.30e9 + 9.18e9i)T^{2} \) |
| 17 | \( 1 + (-2.80e5 + 4.86e5i)T + (-5.92e10 - 1.02e11i)T^{2} \) |
| 23 | \( 1 + (7.61e5 + 1.31e6i)T + (-9.00e11 + 1.55e12i)T^{2} \) |
| 29 | \( 1 + (-1.49e6 - 2.58e6i)T + (-7.25e12 + 1.25e13i)T^{2} \) |
| 31 | \( 1 + 6.51e6T + 2.64e13T^{2} \) |
| 37 | \( 1 + 1.39e7T + 1.29e14T^{2} \) |
| 41 | \( 1 + (5.35e6 - 9.28e6i)T + (-1.63e14 - 2.83e14i)T^{2} \) |
| 43 | \( 1 + (3.93e6 - 6.81e6i)T + (-2.51e14 - 4.35e14i)T^{2} \) |
| 47 | \( 1 + (1.68e7 + 2.91e7i)T + (-5.59e14 + 9.69e14i)T^{2} \) |
| 53 | \( 1 + (8.16e6 + 1.41e7i)T + (-1.64e15 + 2.85e15i)T^{2} \) |
| 59 | \( 1 + (-8.47e6 + 1.46e7i)T + (-4.33e15 - 7.50e15i)T^{2} \) |
| 61 | \( 1 + (-4.50e7 - 7.80e7i)T + (-5.84e15 + 1.01e16i)T^{2} \) |
| 67 | \( 1 + (7.42e7 + 1.28e8i)T + (-1.36e16 + 2.35e16i)T^{2} \) |
| 71 | \( 1 + (6.16e7 - 1.06e8i)T + (-2.29e16 - 3.97e16i)T^{2} \) |
| 73 | \( 1 + (-1.82e8 + 3.16e8i)T + (-2.94e16 - 5.09e16i)T^{2} \) |
| 79 | \( 1 + (1.62e8 - 2.82e8i)T + (-5.99e16 - 1.03e17i)T^{2} \) |
| 83 | \( 1 - 2.17e8T + 1.86e17T^{2} \) |
| 89 | \( 1 + (4.90e8 + 8.49e8i)T + (-1.75e17 + 3.03e17i)T^{2} \) |
| 97 | \( 1 + (-8.55e8 + 1.48e9i)T + (-3.80e17 - 6.58e17i)T^{2} \) |
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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
−16.52056680259842845468299101733, −14.63297388812290717828779514689, −14.12445108874826829166096504490, −12.16770700053417123702389575081, −11.47877661679128067723438306803, −8.629639332503373635467997498439, −7.56810186694219527743098196185, −6.80143550832182348065759425723, −3.33940134829601799594054611474, −1.69356357541565424288453038185,
1.42414357433129571705217124457, 3.84464161477854805863377486634, 5.25443105832559301785003064715, 8.246062360045690456700495329209, 9.315353596907565482127371205203, 10.67483786234802411056242726329, 11.84882032416247216288876050772, 14.23878288146123004639855676131, 15.05832139404337378541069757107, 15.97015287220114458421583606799
