| L(s) = 1 | + (−13.2 − 2.34i)2-s + (−30.7 + 36.6i)3-s + (110. + 40.3i)4-s + (−169. + 61.7i)5-s + (494. − 415. i)6-s + (−29.1 − 50.5i)7-s + (−631. − 364. i)8-s + (−271. − 1.53e3i)9-s + (2.39e3 − 422. i)10-s + (−267. + 462. i)11-s + (−4.89e3 + 2.82e3i)12-s + (2.28e3 + 2.72e3i)13-s + (269. + 739. i)14-s + (2.95e3 − 8.11e3i)15-s + (1.74e3 + 1.46e3i)16-s + (−230. + 1.30e3i)17-s + ⋯ |
| L(s) = 1 | + (−1.66 − 0.292i)2-s + (−1.13 + 1.35i)3-s + (1.73 + 0.630i)4-s + (−1.35 + 0.493i)5-s + (2.28 − 1.92i)6-s + (−0.0850 − 0.147i)7-s + (−1.23 − 0.711i)8-s + (−0.371 − 2.10i)9-s + (2.39 − 0.422i)10-s + (−0.200 + 0.347i)11-s + (−2.83 + 1.63i)12-s + (1.03 + 1.23i)13-s + (0.0980 + 0.269i)14-s + (0.875 − 2.40i)15-s + (0.426 + 0.357i)16-s + (−0.0469 + 0.266i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.310 + 0.950i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.0479841 - 0.0347996i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.0479841 - 0.0347996i\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (3.36e3 + 5.97e3i)T \) |
| good | 2 | \( 1 + (13.2 + 2.34i)T + (60.1 + 21.8i)T^{2} \) |
| 3 | \( 1 + (30.7 - 36.6i)T + (-126. - 717. i)T^{2} \) |
| 5 | \( 1 + (169. - 61.7i)T + (1.19e4 - 1.00e4i)T^{2} \) |
| 7 | \( 1 + (29.1 + 50.5i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (267. - 462. i)T + (-8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-2.28e3 - 2.72e3i)T + (-8.38e5 + 4.75e6i)T^{2} \) |
| 17 | \( 1 + (230. - 1.30e3i)T + (-2.26e7 - 8.25e6i)T^{2} \) |
| 23 | \( 1 + (5.71e3 + 2.08e3i)T + (1.13e8 + 9.51e7i)T^{2} \) |
| 29 | \( 1 + (1.18e4 - 2.09e3i)T + (5.58e8 - 2.03e8i)T^{2} \) |
| 31 | \( 1 + (-1.48e4 + 8.58e3i)T + (4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 - 1.73e4iT - 2.56e9T^{2} \) |
| 41 | \( 1 + (-5.47e4 + 6.53e4i)T + (-8.24e8 - 4.67e9i)T^{2} \) |
| 43 | \( 1 + (8.36e4 - 3.04e4i)T + (4.84e9 - 4.06e9i)T^{2} \) |
| 47 | \( 1 + (2.12e4 + 1.20e5i)T + (-1.01e10 + 3.68e9i)T^{2} \) |
| 53 | \( 1 + (-8.76e3 + 2.40e4i)T + (-1.69e10 - 1.42e10i)T^{2} \) |
| 59 | \( 1 + (3.23e4 + 5.70e3i)T + (3.96e10 + 1.44e10i)T^{2} \) |
| 61 | \( 1 + (-2.51e5 - 9.15e4i)T + (3.94e10 + 3.31e10i)T^{2} \) |
| 67 | \( 1 + (1.25e5 - 2.20e4i)T + (8.50e10 - 3.09e10i)T^{2} \) |
| 71 | \( 1 + (-1.12e5 - 3.08e5i)T + (-9.81e10 + 8.23e10i)T^{2} \) |
| 73 | \( 1 + (2.40e5 + 2.01e5i)T + (2.62e10 + 1.49e11i)T^{2} \) |
| 79 | \( 1 + (4.78e5 - 5.69e5i)T + (-4.22e10 - 2.39e11i)T^{2} \) |
| 83 | \( 1 + (1.42e5 + 2.47e5i)T + (-1.63e11 + 2.83e11i)T^{2} \) |
| 89 | \( 1 + (-3.27e4 - 3.90e4i)T + (-8.62e10 + 4.89e11i)T^{2} \) |
| 97 | \( 1 + (4.11e5 + 7.25e4i)T + (7.82e11 + 2.84e11i)T^{2} \) |
| 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
−16.86081886388301861870500626705, −16.05560228433168215886602265316, −15.25966867396576088601354192666, −11.68430140810303010718711563725, −11.16678465269306283069712739294, −10.12117847061802376917103271285, −8.693692003554539115949670052363, −6.82836866118485798994162434065, −4.06296897323630487212192756969, −0.098182230320172808604681300276,
1.03240137293510210547247935227, 6.07041273671624790450249558042, 7.64708943764612883168395804631, 8.302386123338864962248740859337, 10.72917255154313423404463148828, 11.70216255182072843122514549860, 12.91005804009015343600084305123, 15.71903399329051202352773242634, 16.48799035558702098760418865004, 17.65171811442115016889260910375
