| L(s) = 1 | + (10.9 − 2.70i)2-s + (−45.8 + 19.0i)3-s + (113. − 59.5i)4-s + (−2.22 + 5.37i)5-s + (−452. + 333. i)6-s + (863. + 863. i)7-s + (1.08e3 − 961. i)8-s + (196. − 196. i)9-s + (−9.88 + 65.0i)10-s + (5.28e3 + 2.19e3i)11-s + (−4.06e3 + 4.88e3i)12-s + (3.60e3 + 8.70e3i)13-s + (1.18e4 + 7.14e3i)14-s − 288. i·15-s + (9.29e3 − 1.34e4i)16-s + 1.16e4i·17-s + ⋯ |
| L(s) = 1 | + (0.970 − 0.239i)2-s + (−0.980 + 0.406i)3-s + (0.885 − 0.465i)4-s + (−0.00795 + 0.0192i)5-s + (−0.855 + 0.629i)6-s + (0.951 + 0.951i)7-s + (0.748 − 0.663i)8-s + (0.0900 − 0.0900i)9-s + (−0.00312 + 0.0205i)10-s + (1.19 + 0.496i)11-s + (−0.679 + 0.815i)12-s + (0.455 + 1.09i)13-s + (1.15 + 0.695i)14-s − 0.0220i·15-s + (0.567 − 0.823i)16-s + 0.574i·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & (0.813 - 0.581i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(2.42765 + 0.778089i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.42765 + 0.778089i\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-10.9 + 2.70i)T \) |
| good | 3 | \( 1 + (45.8 - 19.0i)T + (1.54e3 - 1.54e3i)T^{2} \) |
| 5 | \( 1 + (2.22 - 5.37i)T + (-5.52e4 - 5.52e4i)T^{2} \) |
| 7 | \( 1 + (-863. - 863. i)T + 8.23e5iT^{2} \) |
| 11 | \( 1 + (-5.28e3 - 2.19e3i)T + (1.37e7 + 1.37e7i)T^{2} \) |
| 13 | \( 1 + (-3.60e3 - 8.70e3i)T + (-4.43e7 + 4.43e7i)T^{2} \) |
| 17 | \( 1 - 1.16e4iT - 4.10e8T^{2} \) |
| 19 | \( 1 + (1.77e4 + 4.27e4i)T + (-6.32e8 + 6.32e8i)T^{2} \) |
| 23 | \( 1 + (-1.72e3 + 1.72e3i)T - 3.40e9iT^{2} \) |
| 29 | \( 1 + (1.36e5 - 5.64e4i)T + (1.21e10 - 1.21e10i)T^{2} \) |
| 31 | \( 1 + 1.49e5T + 2.75e10T^{2} \) |
| 37 | \( 1 + (-1.31e5 + 3.17e5i)T + (-6.71e10 - 6.71e10i)T^{2} \) |
| 41 | \( 1 + (9.71e4 - 9.71e4i)T - 1.94e11iT^{2} \) |
| 43 | \( 1 + (-1.05e4 - 4.35e3i)T + (1.92e11 + 1.92e11i)T^{2} \) |
| 47 | \( 1 - 1.36e5iT - 5.06e11T^{2} \) |
| 53 | \( 1 + (1.63e3 + 677. i)T + (8.30e11 + 8.30e11i)T^{2} \) |
| 59 | \( 1 + (-4.33e5 + 1.04e6i)T + (-1.75e12 - 1.75e12i)T^{2} \) |
| 61 | \( 1 + (4.54e5 - 1.88e5i)T + (2.22e12 - 2.22e12i)T^{2} \) |
| 67 | \( 1 + (-3.69e6 + 1.53e6i)T + (4.28e12 - 4.28e12i)T^{2} \) |
| 71 | \( 1 + (-3.74e6 - 3.74e6i)T + 9.09e12iT^{2} \) |
| 73 | \( 1 + (1.69e6 - 1.69e6i)T - 1.10e13iT^{2} \) |
| 79 | \( 1 + 7.68e6iT - 1.92e13T^{2} \) |
| 83 | \( 1 + (2.83e6 + 6.83e6i)T + (-1.91e13 + 1.91e13i)T^{2} \) |
| 89 | \( 1 + (1.48e6 + 1.48e6i)T + 4.42e13iT^{2} \) |
| 97 | \( 1 + 1.61e7T + 8.07e13T^{2} \) |
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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
−15.15573714804306799348715231638, −14.35697456872837875634521184207, −12.68442286336778740778240840897, −11.43439419563495959654711459874, −11.11729027324292146770442441224, −9.092260859244138123197681135007, −6.72970171025337111029901405325, −5.42195576852148195539036532731, −4.28674665104536179955448338851, −1.84056161071059552489912184967,
1.15645327673304483039501327179, 3.86069545086664877182684641450, 5.47880444987305709693491095329, 6.64753134978158234040741916546, 8.065058452670903562487084666998, 10.73395517381166121850537842517, 11.54282919260993719437891282974, 12.63192718999701974121960093553, 13.96471486915734855353373178619, 14.86068424651687481538968636950
