| L(s) = 1 | + (−1.33 + 0.459i)2-s + (−0.581 + 0.224i)3-s + (1.57 − 1.22i)4-s + (2.00 − 2.56i)5-s + (0.674 − 0.567i)6-s + (−0.273 + 0.765i)7-s + (−1.54 + 2.37i)8-s + (−1.93 + 1.75i)9-s + (−1.49 + 4.34i)10-s + (0.832 + 3.70i)11-s + (−0.641 + 1.06i)12-s + (3.04 − 0.842i)13-s + (0.0143 − 1.14i)14-s + (−0.588 + 1.93i)15-s + (0.974 − 3.87i)16-s + (1.68 − 0.509i)17-s + ⋯ |
| L(s) = 1 | + (−0.945 + 0.325i)2-s + (−0.335 + 0.129i)3-s + (0.788 − 0.614i)4-s + (0.894 − 1.14i)5-s + (0.275 − 0.231i)6-s + (−0.103 + 0.289i)7-s + (−0.545 + 0.837i)8-s + (−0.645 + 0.584i)9-s + (−0.473 + 1.37i)10-s + (0.250 + 1.11i)11-s + (−0.185 + 0.308i)12-s + (0.844 − 0.233i)13-s + (0.00382 − 0.307i)14-s + (−0.151 + 0.500i)15-s + (0.243 − 0.969i)16-s + (0.407 − 0.123i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.950 - 0.312i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.975297 + 0.156154i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.975297 + 0.156154i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.33 - 0.459i)T \) |
| good | 3 | \( 1 + (0.581 - 0.224i)T + (2.22 - 2.01i)T^{2} \) |
| 5 | \( 1 + (-2.00 + 2.56i)T + (-1.21 - 4.85i)T^{2} \) |
| 7 | \( 1 + (0.273 - 0.765i)T + (-5.41 - 4.44i)T^{2} \) |
| 11 | \( 1 + (-0.832 - 3.70i)T + (-9.94 + 4.70i)T^{2} \) |
| 13 | \( 1 + (-3.04 + 0.842i)T + (11.1 - 6.68i)T^{2} \) |
| 17 | \( 1 + (-1.68 + 0.509i)T + (14.1 - 9.44i)T^{2} \) |
| 19 | \( 1 + (-1.87 - 3.72i)T + (-11.3 + 15.2i)T^{2} \) |
| 23 | \( 1 + (-1.75 + 0.259i)T + (22.0 - 6.67i)T^{2} \) |
| 29 | \( 1 + (0.432 + 2.49i)T + (-27.3 + 9.76i)T^{2} \) |
| 31 | \( 1 + (-4.30 + 6.44i)T + (-11.8 - 28.6i)T^{2} \) |
| 37 | \( 1 + (-6.50 + 5.60i)T + (5.42 - 36.5i)T^{2} \) |
| 41 | \( 1 + (0.197 - 0.790i)T + (-36.1 - 19.3i)T^{2} \) |
| 43 | \( 1 + (-3.21 - 7.25i)T + (-28.8 + 31.8i)T^{2} \) |
| 47 | \( 1 + (-1.10 - 1.35i)T + (-9.16 + 46.0i)T^{2} \) |
| 53 | \( 1 + (-0.193 + 1.11i)T + (-49.9 - 17.8i)T^{2} \) |
| 59 | \( 1 + (1.94 - 7.04i)T + (-50.6 - 30.3i)T^{2} \) |
| 61 | \( 1 + (3.78 + 0.0930i)T + (60.9 + 2.99i)T^{2} \) |
| 67 | \( 1 + (-3.22 + 3.07i)T + (3.28 - 66.9i)T^{2} \) |
| 71 | \( 1 + (4.29 - 0.211i)T + (70.6 - 6.95i)T^{2} \) |
| 73 | \( 1 + (0.404 + 1.13i)T + (-56.4 + 46.3i)T^{2} \) |
| 79 | \( 1 + (-3.95 + 0.389i)T + (77.4 - 15.4i)T^{2} \) |
| 83 | \( 1 + (3.68 + 3.17i)T + (12.1 + 82.1i)T^{2} \) |
| 89 | \( 1 + (7.98 + 1.18i)T + (85.1 + 25.8i)T^{2} \) |
| 97 | \( 1 + (-5.98 + 1.19i)T + (89.6 - 37.1i)T^{2} \) |
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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
−10.72690676309449104826388174307, −9.756651510569628664148628685363, −9.313570856374375369333847180169, −8.343863349367406143813641526106, −7.58576266533850999229614220695, −6.04538900627229549129472476205, −5.71317243100408937305015738013, −4.59007288894945578744852976967, −2.43920411217251471283472733062, −1.19610892076799095645194660229,
1.06392353039803144275877737402, 2.78574450094869953959675808959, 3.48576151482442281323147728690, 5.70301155712544643094412909150, 6.46505096572181815671930709289, 7.02760665532906811074593583669, 8.421551684678303186294370305950, 9.122863702527670874702107927729, 10.06769370048539041623066506932, 10.90835547596894826549132190359
