| L(s) = 1 | + (−0.163 + 0.928i)2-s + (1.56 + 1.30i)3-s + (6.68 + 2.43i)4-s + (−3.55 + 1.29i)5-s + (−1.47 + 1.23i)6-s + (−11.7 − 20.2i)7-s + (−7.12 + 12.3i)8-s + (−3.96 − 22.5i)9-s + (−0.620 − 3.51i)10-s + (−8.50 + 14.7i)11-s + (7.24 + 12.5i)12-s + (3.66 − 3.07i)13-s + (20.7 − 7.55i)14-s + (−7.24 − 2.63i)15-s + (33.2 + 27.9i)16-s + (−9.73 + 55.2i)17-s + ⋯ |
| L(s) = 1 | + (−0.0579 + 0.328i)2-s + (0.300 + 0.252i)3-s + (0.835 + 0.303i)4-s + (−0.318 + 0.115i)5-s + (−0.100 + 0.0840i)6-s + (−0.632 − 1.09i)7-s + (−0.314 + 0.545i)8-s + (−0.146 − 0.833i)9-s + (−0.0196 − 0.111i)10-s + (−0.233 + 0.403i)11-s + (0.174 + 0.301i)12-s + (0.0780 − 0.0655i)13-s + (0.396 − 0.144i)14-s + (−0.124 − 0.0454i)15-s + (0.519 + 0.436i)16-s + (−0.138 + 0.787i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 19 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(1.11980 + 0.326182i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.11980 + 0.326182i\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 19 | \( 1 + (-80.5 + 19.1i)T \) |
| good | 2 | \( 1 + (0.163 - 0.928i)T + (-7.51 - 2.73i)T^{2} \) |
| 3 | \( 1 + (-1.56 - 1.30i)T + (4.68 + 26.5i)T^{2} \) |
| 5 | \( 1 + (3.55 - 1.29i)T + (95.7 - 80.3i)T^{2} \) |
| 7 | \( 1 + (11.7 + 20.2i)T + (-171.5 + 297. i)T^{2} \) |
| 11 | \( 1 + (8.50 - 14.7i)T + (-665.5 - 1.15e3i)T^{2} \) |
| 13 | \( 1 + (-3.66 + 3.07i)T + (381. - 2.16e3i)T^{2} \) |
| 17 | \( 1 + (9.73 - 55.2i)T + (-4.61e3 - 1.68e3i)T^{2} \) |
| 23 | \( 1 + (83.8 + 30.5i)T + (9.32e3 + 7.82e3i)T^{2} \) |
| 29 | \( 1 + (-29.8 - 169. i)T + (-2.29e4 + 8.34e3i)T^{2} \) |
| 31 | \( 1 + (48.1 + 83.4i)T + (-1.48e4 + 2.57e4i)T^{2} \) |
| 37 | \( 1 - 339.T + 5.06e4T^{2} \) |
| 41 | \( 1 + (-339. - 285. i)T + (1.19e4 + 6.78e4i)T^{2} \) |
| 43 | \( 1 + (253. - 92.2i)T + (6.09e4 - 5.11e4i)T^{2} \) |
| 47 | \( 1 + (84.2 + 477. i)T + (-9.75e4 + 3.55e4i)T^{2} \) |
| 53 | \( 1 + (594. + 216. i)T + (1.14e5 + 9.56e4i)T^{2} \) |
| 59 | \( 1 + (-114. + 648. i)T + (-1.92e5 - 7.02e4i)T^{2} \) |
| 61 | \( 1 + (135. + 49.4i)T + (1.73e5 + 1.45e5i)T^{2} \) |
| 67 | \( 1 + (-62.4 - 354. i)T + (-2.82e5 + 1.02e5i)T^{2} \) |
| 71 | \( 1 + (-998. + 363. i)T + (2.74e5 - 2.30e5i)T^{2} \) |
| 73 | \( 1 + (-227. - 190. i)T + (6.75e4 + 3.83e5i)T^{2} \) |
| 79 | \( 1 + (81.3 + 68.2i)T + (8.56e4 + 4.85e5i)T^{2} \) |
| 83 | \( 1 + (332. + 576. i)T + (-2.85e5 + 4.95e5i)T^{2} \) |
| 89 | \( 1 + (252. - 212. i)T + (1.22e5 - 6.94e5i)T^{2} \) |
| 97 | \( 1 + (-4.50 + 25.5i)T + (-8.57e5 - 3.12e5i)T^{2} \) |
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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
−17.91940482334611140628447918542, −16.66868446440154819942606828868, −15.66515301318324257238255302383, −14.55648937704823636597185395401, −12.83487513232693792519773163910, −11.33880121378812531855040718215, −9.819374711829168073340396054837, −7.83786963101891547203564025649, −6.51226115841555254826716575508, −3.54438302940566114172588536959,
2.63565063763629509856029408625, 5.88137772130247378957566603365, 7.78914084896758339681067899936, 9.594970643934432887604441071906, 11.27461169876256544661257160359, 12.36464931104606340028961490913, 13.94608201618549319900996910397, 15.65017650543303117275900626676, 16.21190588616418465536235188593, 18.38068390973867927170272990646
