| L(s) = 1 | + (4 − 6.92i)2-s + (7.15 − 12.3i)3-s + (−31.9 − 55.4i)4-s + (−222. + 385. i)5-s + (−57.2 − 99.1i)6-s + 1.43e3·7-s − 511.·8-s + (991. + 1.71e3i)9-s + (1.78e3 + 3.08e3i)10-s + 5.16e3·11-s − 915.·12-s + (−4.98e3 − 8.63e3i)13-s + (5.74e3 − 9.94e3i)14-s + (3.18e3 + 5.51e3i)15-s + (−2.04e3 + 3.54e3i)16-s + (−5.59e3 + 9.68e3i)17-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (0.153 − 0.265i)3-s + (−0.249 − 0.433i)4-s + (−0.796 + 1.37i)5-s + (−0.108 − 0.187i)6-s + 1.58·7-s − 0.353·8-s + (0.453 + 0.784i)9-s + (0.563 + 0.975i)10-s + 1.16·11-s − 0.153·12-s + (−0.629 − 1.09i)13-s + (0.559 − 0.969i)14-s + (0.243 + 0.422i)15-s + (−0.125 + 0.216i)16-s + (−0.276 + 0.478i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 38 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0463i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.30939 - 0.0535568i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.30939 - 0.0535568i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-4 + 6.92i)T \) |
| 19 | \( 1 + (-2.88e4 - 7.66e3i)T \) |
| good | 3 | \( 1 + (-7.15 + 12.3i)T + (-1.09e3 - 1.89e3i)T^{2} \) |
| 5 | \( 1 + (222. - 385. i)T + (-3.90e4 - 6.76e4i)T^{2} \) |
| 7 | \( 1 - 1.43e3T + 8.23e5T^{2} \) |
| 11 | \( 1 - 5.16e3T + 1.94e7T^{2} \) |
| 13 | \( 1 + (4.98e3 + 8.63e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + (5.59e3 - 9.68e3i)T + (-2.05e8 - 3.55e8i)T^{2} \) |
| 23 | \( 1 + (-3.34e4 - 5.78e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-5.89e4 - 1.02e5i)T + (-8.62e9 + 1.49e10i)T^{2} \) |
| 31 | \( 1 - 1.54e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + 4.78e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (1.78e5 - 3.08e5i)T + (-9.73e10 - 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.75e4 - 3.03e4i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (5.74e5 + 9.94e5i)T + (-2.53e11 + 4.38e11i)T^{2} \) |
| 53 | \( 1 + (-1.74e5 - 3.01e5i)T + (-5.87e11 + 1.01e12i)T^{2} \) |
| 59 | \( 1 + (-5.09e5 + 8.82e5i)T + (-1.24e12 - 2.15e12i)T^{2} \) |
| 61 | \( 1 + (9.24e5 + 1.60e6i)T + (-1.57e12 + 2.72e12i)T^{2} \) |
| 67 | \( 1 + (1.34e6 + 2.33e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + (-1.51e6 + 2.62e6i)T + (-4.54e12 - 7.87e12i)T^{2} \) |
| 73 | \( 1 + (2.86e6 - 4.96e6i)T + (-5.52e12 - 9.56e12i)T^{2} \) |
| 79 | \( 1 + (-2.37e6 + 4.10e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 - 2.86e6T + 2.71e13T^{2} \) |
| 89 | \( 1 + (5.58e6 + 9.67e6i)T + (-2.21e13 + 3.83e13i)T^{2} \) |
| 97 | \( 1 + (3.49e6 - 6.05e6i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
| 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
−14.62938653020005790125379269148, −13.79300565587211002055503399576, −12.04750501619627718752203205427, −11.21891135016877806737198536934, −10.29571580734208187167682834243, −8.147816757155324563552896162836, −7.10187611796924205446916670932, −4.93732910183986903532238306897, −3.31185577405350268675449380372, −1.60327645031020513536435969366,
1.09782176147166385029670770638, 4.24374740158509935866302506001, 4.82817869085080340466794061981, 7.03316127475864092848670789230, 8.446760676377081977812201499614, 9.252897501573561307868907830813, 11.73642139701582679002574832812, 12.12457628727027159311972330840, 13.90754449186255122759912747996, 14.83897102939003318596545102770
