| L(s) = 1 | + (−4 + 6.92i)2-s + (−31.9 − 55.4i)4-s + (82.5 + 142. i)5-s + (254 − 439. i)7-s + 511.·8-s − 1.32e3·10-s + (−1.51e3 + 2.61e3i)11-s + (−2.51e3 − 4.36e3i)13-s + (2.03e3 + 3.51e3i)14-s + (−2.04e3 + 3.54e3i)16-s − 3.18e3·17-s + 1.50e3·19-s + (5.28e3 − 9.14e3i)20-s + (−1.20e4 − 2.09e4i)22-s + (3.78e4 + 6.54e4i)23-s + ⋯ |
| L(s) = 1 | + (−0.353 + 0.612i)2-s + (−0.249 − 0.433i)4-s + (0.295 + 0.511i)5-s + (0.279 − 0.484i)7-s + 0.353·8-s − 0.417·10-s + (−0.342 + 0.593i)11-s + (−0.318 − 0.550i)13-s + (0.197 + 0.342i)14-s + (−0.125 + 0.216i)16-s − 0.157·17-s + 0.0504·19-s + (0.147 − 0.255i)20-s + (−0.242 − 0.419i)22-s + (0.647 + 1.12i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (-0.766 - 0.642i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.187156796\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187156796\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (4 - 6.92i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (-82.5 - 142. i)T + (-3.90e4 + 6.76e4i)T^{2} \) |
| 7 | \( 1 + (-254 + 439. i)T + (-4.11e5 - 7.13e5i)T^{2} \) |
| 11 | \( 1 + (1.51e3 - 2.61e3i)T + (-9.74e6 - 1.68e7i)T^{2} \) |
| 13 | \( 1 + (2.51e3 + 4.36e3i)T + (-3.13e7 + 5.43e7i)T^{2} \) |
| 17 | \( 1 + 3.18e3T + 4.10e8T^{2} \) |
| 19 | \( 1 - 1.50e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + (-3.78e4 - 6.54e4i)T + (-1.70e9 + 2.94e9i)T^{2} \) |
| 29 | \( 1 + (-4.13e4 + 7.15e4i)T + (-8.62e9 - 1.49e10i)T^{2} \) |
| 31 | \( 1 + (-8.74e4 - 1.51e5i)T + (-1.37e10 + 2.38e10i)T^{2} \) |
| 37 | \( 1 + 3.23e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + (-1.54e5 - 2.66e5i)T + (-9.73e10 + 1.68e11i)T^{2} \) |
| 43 | \( 1 + (1.68e5 - 2.91e5i)T + (-1.35e11 - 2.35e11i)T^{2} \) |
| 47 | \( 1 + (-1.91e5 + 3.31e5i)T + (-2.53e11 - 4.38e11i)T^{2} \) |
| 53 | \( 1 - 7.60e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + (-1.11e6 - 1.92e6i)T + (-1.24e12 + 2.15e12i)T^{2} \) |
| 61 | \( 1 + (1.12e6 - 1.94e6i)T + (-1.57e12 - 2.72e12i)T^{2} \) |
| 67 | \( 1 + (7.36e5 + 1.27e6i)T + (-3.03e12 + 5.24e12i)T^{2} \) |
| 71 | \( 1 + 5.00e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 5.89e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + (3.51e6 - 6.08e6i)T + (-9.60e12 - 1.66e13i)T^{2} \) |
| 83 | \( 1 + (-1.32e6 + 2.29e6i)T + (-1.35e13 - 2.35e13i)T^{2} \) |
| 89 | \( 1 + 6.77e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + (8.08e6 - 1.40e7i)T + (-4.03e13 - 6.99e13i)T^{2} \) |
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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
−11.86294178423367553101231724528, −10.56607809836380438973132556600, −10.04121683077560960670804518646, −8.785784853223215830564356780938, −7.61509864994735350430822832948, −6.89607523530378035248036911493, −5.62531752091792471290518769744, −4.48507694899730310043422621959, −2.79444027874867279335848791864, −1.21458739245951722063768984638,
0.37923359679752009997917289257, 1.73330536103609090636194954069, 2.92596122790365499405313959050, 4.49389301607599018444306885408, 5.57663560894673460425593932889, 7.06154772322515947533553879971, 8.481410257141197917208365841276, 9.006272271154885158694038166780, 10.18586857157900668119663358340, 11.16012619144602214346108691040
