| L(s) = 1 | + (−3.41 − 4.06i)2-s + (−4.67 + 12.8i)3-s + (−2.10 + 11.9i)4-s + (3.16 + 17.9i)5-s + (68.1 − 24.7i)6-s + (−22.5 + 39.0i)7-s + (−17.7 + 10.2i)8-s + (−80.9 − 67.8i)9-s + (62.1 − 74.0i)10-s + (−70.6 − 122. i)11-s + (−143. − 82.9i)12-s + (78.7 + 216. i)13-s + (235. − 41.5i)14-s + (−245. − 43.2i)15-s + (284. + 103. i)16-s + (382. − 320. i)17-s + ⋯ |
| L(s) = 1 | + (−0.852 − 1.01i)2-s + (−0.519 + 1.42i)3-s + (−0.131 + 0.747i)4-s + (0.126 + 0.717i)5-s + (1.89 − 0.688i)6-s + (−0.460 + 0.797i)7-s + (−0.276 + 0.159i)8-s + (−0.998 − 0.838i)9-s + (0.621 − 0.740i)10-s + (−0.584 − 1.01i)11-s + (−0.997 − 0.576i)12-s + (0.465 + 1.27i)13-s + (1.20 − 0.212i)14-s + (−1.08 − 0.192i)15-s + (1.11 + 0.404i)16-s + (1.32 − 1.11i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0212 - 0.999i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.0212 - 0.999i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.377480 + 0.369555i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.377480 + 0.369555i\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (237. - 271. i)T \) |
| good | 2 | \( 1 + (3.41 + 4.06i)T + (-2.77 + 15.7i)T^{2} \) |
| 3 | \( 1 + (4.67 - 12.8i)T + (-62.0 - 52.0i)T^{2} \) |
| 5 | \( 1 + (-3.16 - 17.9i)T + (-587. + 213. i)T^{2} \) |
| 7 | \( 1 + (22.5 - 39.0i)T + (-1.20e3 - 2.07e3i)T^{2} \) |
| 11 | \( 1 + (70.6 + 122. i)T + (-7.32e3 + 1.26e4i)T^{2} \) |
| 13 | \( 1 + (-78.7 - 216. i)T + (-2.18e4 + 1.83e4i)T^{2} \) |
| 17 | \( 1 + (-382. + 320. i)T + (1.45e4 - 8.22e4i)T^{2} \) |
| 23 | \( 1 + (52.8 - 299. i)T + (-2.62e5 - 9.57e4i)T^{2} \) |
| 29 | \( 1 + (487. - 581. i)T + (-1.22e5 - 6.96e5i)T^{2} \) |
| 31 | \( 1 + (-662. - 382. i)T + (4.61e5 + 7.99e5i)T^{2} \) |
| 37 | \( 1 - 660. iT - 1.87e6T^{2} \) |
| 41 | \( 1 + (-74.5 + 204. i)T + (-2.16e6 - 1.81e6i)T^{2} \) |
| 43 | \( 1 + (197. + 1.11e3i)T + (-3.21e6 + 1.16e6i)T^{2} \) |
| 47 | \( 1 + (-1.56e3 - 1.31e3i)T + (8.47e5 + 4.80e6i)T^{2} \) |
| 53 | \( 1 + (-48.9 - 8.63i)T + (7.41e6 + 2.69e6i)T^{2} \) |
| 59 | \( 1 + (-835. - 995. i)T + (-2.10e6 + 1.19e7i)T^{2} \) |
| 61 | \( 1 + (-547. + 3.10e3i)T + (-1.30e7 - 4.73e6i)T^{2} \) |
| 67 | \( 1 + (2.27e3 - 2.70e3i)T + (-3.49e6 - 1.98e7i)T^{2} \) |
| 71 | \( 1 + (-437. + 77.1i)T + (2.38e7 - 8.69e6i)T^{2} \) |
| 73 | \( 1 + (2.87e3 + 1.04e3i)T + (2.17e7 + 1.82e7i)T^{2} \) |
| 79 | \( 1 + (-839. + 2.30e3i)T + (-2.98e7 - 2.50e7i)T^{2} \) |
| 83 | \( 1 + (4.34e3 - 7.52e3i)T + (-2.37e7 - 4.11e7i)T^{2} \) |
| 89 | \( 1 + (5.26e3 + 1.44e4i)T + (-4.80e7 + 4.03e7i)T^{2} \) |
| 97 | \( 1 + (-9.70e3 - 1.15e4i)T + (-1.53e7 + 8.71e7i)T^{2} \) |
| 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
−18.55588709662091751753750798215, −16.76376512446105895179146237606, −15.81961446957459259923962365309, −14.36840824388610008378079949218, −11.91414493843971930238482964032, −10.92973175215638834920802000300, −9.973097498540398207766302978518, −8.913941426010533925983541001559, −5.76682396970592726152207649409, −3.18345527994155426308719873963,
0.68277207251816821846089228065, 5.91079459636583595097087193271, 7.27022458693342442189064745950, 8.224235655827781676502682353124, 10.19953278393243514174464015582, 12.53534756409350916956655578242, 13.08395012843320771297199054041, 15.18728721179834727459141505729, 16.72512719758513845918608903225, 17.35686223187141367496648564038
