| L(s) = 1 | + (−4.89 + 2.82i)2-s + (15.9 − 27.7i)4-s + (117. + 68.0i)5-s + (93.5 + 161. i)7-s + 181. i·8-s − 769.·10-s + (−1.59e3 + 920. i)11-s + (290. − 503. i)13-s + (−916. − 529. i)14-s + (−512. − 886. i)16-s + 1.75e3i·17-s − 7.71e3·19-s + (3.77e3 − 2.17e3i)20-s + (5.20e3 − 9.01e3i)22-s + (1.27e4 + 7.35e3i)23-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (0.249 − 0.433i)4-s + (0.942 + 0.544i)5-s + (0.272 + 0.472i)7-s + 0.353i·8-s − 0.769·10-s + (−1.19 + 0.691i)11-s + (0.132 − 0.229i)13-s + (−0.333 − 0.192i)14-s + (−0.125 − 0.216i)16-s + 0.357i·17-s − 1.12·19-s + (0.471 − 0.272i)20-s + (0.489 − 0.847i)22-s + (1.04 + 0.604i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (-0.996 - 0.0871i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{7}{2})\) |
\(\approx\) |
\(0.8293335268\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8293335268\) |
| \(L(4)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4.89 - 2.82i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-117. - 68.0i)T + (7.81e3 + 1.35e4i)T^{2} \) |
| 7 | \( 1 + (-93.5 - 161. i)T + (-5.88e4 + 1.01e5i)T^{2} \) |
| 11 | \( 1 + (1.59e3 - 920. i)T + (8.85e5 - 1.53e6i)T^{2} \) |
| 13 | \( 1 + (-290. + 503. i)T + (-2.41e6 - 4.18e6i)T^{2} \) |
| 17 | \( 1 - 1.75e3iT - 2.41e7T^{2} \) |
| 19 | \( 1 + 7.71e3T + 4.70e7T^{2} \) |
| 23 | \( 1 + (-1.27e4 - 7.35e3i)T + (7.40e7 + 1.28e8i)T^{2} \) |
| 29 | \( 1 + (-2.76e4 + 1.59e4i)T + (2.97e8 - 5.15e8i)T^{2} \) |
| 31 | \( 1 + (1.70e4 - 2.94e4i)T + (-4.43e8 - 7.68e8i)T^{2} \) |
| 37 | \( 1 + 7.70e4T + 2.56e9T^{2} \) |
| 41 | \( 1 + (9.83e3 + 5.67e3i)T + (2.37e9 + 4.11e9i)T^{2} \) |
| 43 | \( 1 + (-2.51e4 - 4.35e4i)T + (-3.16e9 + 5.47e9i)T^{2} \) |
| 47 | \( 1 + (-2.24e4 + 1.29e4i)T + (5.38e9 - 9.33e9i)T^{2} \) |
| 53 | \( 1 - 1.95e5iT - 2.21e10T^{2} \) |
| 59 | \( 1 + (3.31e5 + 1.91e5i)T + (2.10e10 + 3.65e10i)T^{2} \) |
| 61 | \( 1 + (2.23e5 + 3.86e5i)T + (-2.57e10 + 4.46e10i)T^{2} \) |
| 67 | \( 1 + (2.32e5 - 4.02e5i)T + (-4.52e10 - 7.83e10i)T^{2} \) |
| 71 | \( 1 - 2.48e5iT - 1.28e11T^{2} \) |
| 73 | \( 1 + 5.45e5T + 1.51e11T^{2} \) |
| 79 | \( 1 + (5.51e4 + 9.54e4i)T + (-1.21e11 + 2.10e11i)T^{2} \) |
| 83 | \( 1 + (4.10e4 - 2.36e4i)T + (1.63e11 - 2.83e11i)T^{2} \) |
| 89 | \( 1 - 3.78e4iT - 4.96e11T^{2} \) |
| 97 | \( 1 + (-4.48e5 - 7.76e5i)T + (-4.16e11 + 7.21e11i)T^{2} \) |
| 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
−12.26039100781327859391521750078, −10.72481949501223264600418740753, −10.33536953653798236283232497315, −9.170345068342198321813770533278, −8.164616450098204925253036826051, −7.00594211617523401155116042379, −5.95905467221713636646299966823, −4.93715522974692492399729516924, −2.75340338276482243319692264875, −1.71228143720571799504071691712,
0.27722908133271016241331928977, 1.59478939391931861009815509811, 2.87783598072244491797394140645, 4.63849666616678681830402211043, 5.84352321557632503499097759408, 7.19377087483399148436169260712, 8.436936948547075641154076531351, 9.147437700684512262887793083187, 10.43458934704372845431768070915, 10.85243173488945051747916816303
