| L(s) = 1 | + (2 − 3.46i)2-s + (−7.99 − 13.8i)4-s + (−33 − 57.1i)5-s + (−88 + 152. i)7-s − 63.9·8-s − 264·10-s + (−30 + 51.9i)11-s + (329 + 569. i)13-s + (352 + 609. i)14-s + (−128 + 221. i)16-s + 414·17-s + 956·19-s + (−528 + 914. i)20-s + (120 + 207. i)22-s + (300 + 519. i)23-s + ⋯ |
| L(s) = 1 | + (0.353 − 0.612i)2-s + (−0.249 − 0.433i)4-s + (−0.590 − 1.02i)5-s + (−0.678 + 1.17i)7-s − 0.353·8-s − 0.834·10-s + (−0.0747 + 0.129i)11-s + (0.539 + 0.935i)13-s + (0.479 + 0.831i)14-s + (−0.125 + 0.216i)16-s + 0.347·17-s + 0.607·19-s + (−0.295 + 0.511i)20-s + (0.0528 + 0.0915i)22-s + (0.118 + 0.204i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 162 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.939 - 0.342i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.446970679\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.446970679\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-2 + 3.46i)T \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + (33 + 57.1i)T + (-1.56e3 + 2.70e3i)T^{2} \) |
| 7 | \( 1 + (88 - 152. i)T + (-8.40e3 - 1.45e4i)T^{2} \) |
| 11 | \( 1 + (30 - 51.9i)T + (-8.05e4 - 1.39e5i)T^{2} \) |
| 13 | \( 1 + (-329 - 569. i)T + (-1.85e5 + 3.21e5i)T^{2} \) |
| 17 | \( 1 - 414T + 1.41e6T^{2} \) |
| 19 | \( 1 - 956T + 2.47e6T^{2} \) |
| 23 | \( 1 + (-300 - 519. i)T + (-3.21e6 + 5.57e6i)T^{2} \) |
| 29 | \( 1 + (-2.78e3 + 4.82e3i)T + (-1.02e7 - 1.77e7i)T^{2} \) |
| 31 | \( 1 + (-1.79e3 - 3.11e3i)T + (-1.43e7 + 2.47e7i)T^{2} \) |
| 37 | \( 1 + 8.45e3T + 6.93e7T^{2} \) |
| 41 | \( 1 + (-9.59e3 - 1.66e4i)T + (-5.79e7 + 1.00e8i)T^{2} \) |
| 43 | \( 1 + (6.65e3 - 1.15e4i)T + (-7.35e7 - 1.27e8i)T^{2} \) |
| 47 | \( 1 + (9.84e3 - 1.70e4i)T + (-1.14e8 - 1.98e8i)T^{2} \) |
| 53 | \( 1 - 3.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + (-1.31e4 - 2.28e4i)T + (-3.57e8 + 6.19e8i)T^{2} \) |
| 61 | \( 1 + (-1.55e4 + 2.69e4i)T + (-4.22e8 - 7.31e8i)T^{2} \) |
| 67 | \( 1 + (-8.40e3 - 1.45e4i)T + (-6.75e8 + 1.16e9i)T^{2} \) |
| 71 | \( 1 + 6.12e3T + 1.80e9T^{2} \) |
| 73 | \( 1 + 2.55e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + (3.72e4 - 6.44e4i)T + (-1.53e9 - 2.66e9i)T^{2} \) |
| 83 | \( 1 + (3.23e3 - 5.60e3i)T + (-1.96e9 - 3.41e9i)T^{2} \) |
| 89 | \( 1 - 3.27e4T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.30e4 - 1.43e5i)T + (-4.29e9 - 7.43e9i)T^{2} \) |
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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
−12.03862313276269817443236552260, −11.44599180217576130406331988773, −9.864808419257020752482010093121, −9.066480256802874639223227844858, −8.183547698941227998986595933953, −6.43993901506786695393637258436, −5.27988647598606773773556913638, −4.17053614048430523978843520804, −2.79572615860706246305250595709, −1.18268506728603258127753662652,
0.49019946487129206942234873435, 3.15687136856707435364339014561, 3.83726253818949210828191344914, 5.50639799335371246321401698355, 6.85952592027834732659941452985, 7.33626192498415116717211737082, 8.529197023935716213472784550273, 10.14450077903738874267086658423, 10.75131981621552018410509488516, 11.98875818054334684521571466455
