| L(s) = 1 | + (−1.74 + 1.46i)2-s + (4.00 + 1.45i)3-s + (−0.485 + 2.75i)4-s + (1.27 + 7.22i)5-s + (−9.13 + 3.32i)6-s + (13.1 − 22.6i)7-s + (−12.3 − 21.3i)8-s + (−6.77 − 5.68i)9-s + (−12.8 − 10.7i)10-s + (7.39 + 12.8i)11-s + (−5.95 + 10.3i)12-s + (22.9 − 8.37i)13-s + (10.3 + 58.8i)14-s + (−5.42 + 30.7i)15-s + (31.7 + 11.5i)16-s + (−25.4 + 21.3i)17-s + ⋯ |
| L(s) = 1 | + (−0.617 + 0.518i)2-s + (0.770 + 0.280i)3-s + (−0.0606 + 0.344i)4-s + (0.113 + 0.646i)5-s + (−0.621 + 0.226i)6-s + (0.707 − 1.22i)7-s + (−0.544 − 0.942i)8-s + (−0.251 − 0.210i)9-s + (−0.405 − 0.340i)10-s + (0.202 + 0.350i)11-s + (−0.143 + 0.248i)12-s + (0.490 − 0.178i)13-s + (0.198 + 1.12i)14-s + (−0.0934 + 0.529i)15-s + (0.496 + 0.180i)16-s + (−0.362 + 0.304i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.465 - 0.885i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.837573 + 0.505902i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.837573 + 0.505902i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (18.8 + 80.6i)T \) |
| good | 2 | \( 1 + (1.74 - 1.46i)T + (1.38 - 7.87i)T^{2} \) |
| 3 | \( 1 + (-4.00 - 1.45i)T + (20.6 + 17.3i)T^{2} \) |
| 5 | \( 1 + (-1.27 - 7.22i)T + (-117. + 42.7i)T^{2} \) |
| 7 | \( 1 + (-13.1 + 22.6i)T + (-171.5 - 297. i)T^{2} \) |
| 11 | \( 1 + (-7.39 - 12.8i)T + (-665.5 + 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-22.9 + 8.37i)T + (1.68e3 - 1.41e3i)T^{2} \) |
| 17 | \( 1 + (25.4 - 21.3i)T + (853. - 4.83e3i)T^{2} \) |
| 23 | \( 1 + (35.7 - 202. i)T + (-1.14e4 - 4.16e3i)T^{2} \) |
| 29 | \( 1 + (142. + 119. i)T + (4.23e3 + 2.40e4i)T^{2} \) |
| 31 | \( 1 + (70.1 - 121. i)T + (-1.48e4 - 2.57e4i)T^{2} \) |
| 37 | \( 1 + 258.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (3.41 + 1.24i)T + (5.27e4 + 4.43e4i)T^{2} \) |
| 43 | \( 1 + (-90.2 - 511. i)T + (-7.47e4 + 2.71e4i)T^{2} \) |
| 47 | \( 1 + (6.20 + 5.20i)T + (1.80e4 + 1.02e5i)T^{2} \) |
| 53 | \( 1 + (-84.3 + 478. i)T + (-1.39e5 - 5.09e4i)T^{2} \) |
| 59 | \( 1 + (-283. + 237. i)T + (3.56e4 - 2.02e5i)T^{2} \) |
| 61 | \( 1 + (-53.5 + 303. i)T + (-2.13e5 - 7.76e4i)T^{2} \) |
| 67 | \( 1 + (-444. - 372. i)T + (5.22e4 + 2.96e5i)T^{2} \) |
| 71 | \( 1 + (-30.3 - 172. i)T + (-3.36e5 + 1.22e5i)T^{2} \) |
| 73 | \( 1 + (-936. - 341. i)T + (2.98e5 + 2.50e5i)T^{2} \) |
| 79 | \( 1 + (616. + 224. i)T + (3.77e5 + 3.16e5i)T^{2} \) |
| 83 | \( 1 + (138. - 239. i)T + (-2.85e5 - 4.95e5i)T^{2} \) |
| 89 | \( 1 + (-731. + 266. i)T + (5.40e5 - 4.53e5i)T^{2} \) |
| 97 | \( 1 + (949. - 796. i)T + (1.58e5 - 8.98e5i)T^{2} \) |
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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
−17.82695670456148101936839545450, −17.30309017670283070418551693212, −15.67406199475263206197708730070, −14.53547520563851118548113002654, −13.34573316714556850706563089641, −11.19868866383369013892206998779, −9.587598555244663240182369253383, −8.192702990535035207106626617073, −6.95986660092469369825546336935, −3.69793844095139992981284266863,
2.06785703035997677879071887368, 5.51354498622450979566295857076, 8.509988622645870089922165479890, 8.930092216837932209266051553056, 10.89817523718342887012272360257, 12.29470448513664124703235729719, 14.03211547125771352535746673149, 14.98387386077710269397123249061, 16.75221695958834663200273548411, 18.40605237869654640100029700088
