| L(s) = 1 | + (−5.03 + 1.83i)2-s + (0.944 − 5.35i)3-s + (15.9 − 13.3i)4-s + (−7.64 − 6.41i)5-s + (5.06 + 28.7i)6-s + (−9.43 − 16.3i)7-s + (−34.2 + 59.2i)8-s + (−2.44 − 0.890i)9-s + (50.3 + 18.3i)10-s + (11.4 − 19.8i)11-s + (−56.4 − 97.8i)12-s + (3.39 + 19.2i)13-s + (77.4 + 65.0i)14-s + (−41.6 + 34.9i)15-s + (34.8 − 197. i)16-s + (−6.05 + 2.20i)17-s + ⋯ |
| L(s) = 1 | + (−1.78 + 0.648i)2-s + (0.181 − 1.03i)3-s + (1.98 − 1.66i)4-s + (−0.683 − 0.573i)5-s + (0.344 + 1.95i)6-s + (−0.509 − 0.881i)7-s + (−1.51 + 2.61i)8-s + (−0.0905 − 0.0329i)9-s + (1.59 + 0.578i)10-s + (0.313 − 0.543i)11-s + (−1.35 − 2.35i)12-s + (0.0724 + 0.410i)13-s + (1.47 + 1.24i)14-s + (−0.716 + 0.600i)15-s + (0.545 − 3.09i)16-s + (−0.0863 + 0.0314i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.136 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.336392 - 0.293071i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.336392 - 0.293071i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-68.3 - 46.7i)T \) |
| good | 2 | \( 1 + (5.03 - 1.83i)T + (6.12 - 5.14i)T^{2} \) |
| 3 | \( 1 + (-0.944 + 5.35i)T + (-25.3 - 9.23i)T^{2} \) |
| 5 | \( 1 + (7.64 + 6.41i)T + (21.7 + 123. i)T^{2} \) |
| 7 | \( 1 + (9.43 + 16.3i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (-11.4 + 19.8i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-3.39 - 19.2i)T + (-2.06e3 + 751. i)T^{2} \) |
| 17 | \( 1 + (6.05 - 2.20i)T + (3.76e3 - 3.15e3i)T^{2} \) |
| 23 | \( 1 + (-77.7 + 65.2i)T + (2.11e3 - 1.19e4i)T^{2} \) |
| 29 | \( 1 + (-96.3 - 35.0i)T + (1.86e4 + 1.56e4i)T^{2} \) |
| 31 | \( 1 + (-43.6 - 75.6i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 + 389.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-35.0 + 198. i)T + (-6.47e4 - 2.35e4i)T^{2} \) |
| 43 | \( 1 + (-289. - 243. i)T + (1.38e4 + 7.82e4i)T^{2} \) |
| 47 | \( 1 + (-120. - 43.6i)T + (7.95e4 + 6.67e4i)T^{2} \) |
| 53 | \( 1 + (-67.0 + 56.2i)T + (2.58e4 - 1.46e5i)T^{2} \) |
| 59 | \( 1 + (243. - 88.6i)T + (1.57e5 - 1.32e5i)T^{2} \) |
| 61 | \( 1 + (385. - 323. i)T + (3.94e4 - 2.23e5i)T^{2} \) |
| 67 | \( 1 + (-601. - 218. i)T + (2.30e5 + 1.93e5i)T^{2} \) |
| 71 | \( 1 + (532. + 446. i)T + (6.21e4 + 3.52e5i)T^{2} \) |
| 73 | \( 1 + (41.0 - 232. i)T + (-3.65e5 - 1.33e5i)T^{2} \) |
| 79 | \( 1 + (-199. + 1.12e3i)T + (-4.63e5 - 1.68e5i)T^{2} \) |
| 83 | \( 1 + (-359. - 623. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-237. - 1.34e3i)T + (-6.62e5 + 2.41e5i)T^{2} \) |
| 97 | \( 1 + (277. - 101. i)T + (6.99e5 - 5.86e5i)T^{2} \) |
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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
−17.81010274209691861022317639988, −16.60436826956017138956734781796, −15.89330238248336965971112712288, −13.98857326662450205168200126583, −12.11598221657473336230814807431, −10.47647793212569175664554868789, −8.835177781091336388757253153170, −7.66194267192290657805029305545, −6.65358536410528724271232728781, −0.937193042432116374902735767072,
3.19140940916957848991751411189, 7.22993683311545288760203943938, 8.940065004468136200102930734579, 9.846099296103752658075437958815, 11.06368382526455484721451268850, 12.25328514234841492277040663284, 15.43836231903177950437772676753, 15.70865307770408502077865188511, 17.28568818069790557672830307627, 18.52701495977298877221119441629
