| L(s) = 1 | + (−1.26 + 0.630i)2-s + (−1.05 − 1.37i)3-s + (1.20 − 1.59i)4-s + (0.00886 + 0.0673i)5-s + (2.20 + 1.07i)6-s + (0.666 + 2.48i)7-s + (−0.519 + 2.78i)8-s + (−0.778 + 2.89i)9-s + (−0.0536 − 0.0796i)10-s + (0.332 − 0.255i)11-s + (−3.46 + 0.0262i)12-s + (3.75 − 4.89i)13-s + (−2.41 − 2.72i)14-s + (0.0831 − 0.0831i)15-s + (−1.09 − 3.84i)16-s + 3.64·17-s + ⋯ |
| L(s) = 1 | + (−0.895 + 0.445i)2-s + (−0.608 − 0.793i)3-s + (0.602 − 0.798i)4-s + (0.00396 + 0.0300i)5-s + (0.898 + 0.439i)6-s + (0.252 + 0.940i)7-s + (−0.183 + 0.983i)8-s + (−0.259 + 0.965i)9-s + (−0.0169 − 0.0251i)10-s + (0.100 − 0.0768i)11-s + (−0.999 + 0.00758i)12-s + (1.04 − 1.35i)13-s + (−0.644 − 0.729i)14-s + (0.0214 − 0.0214i)15-s + (−0.273 − 0.961i)16-s + 0.885·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 288 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.959 + 0.282i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.752076 - 0.108419i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.752076 - 0.108419i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.26 - 0.630i)T \) |
| 3 | \( 1 + (1.05 + 1.37i)T \) |
| good | 5 | \( 1 + (-0.00886 - 0.0673i)T + (-4.82 + 1.29i)T^{2} \) |
| 7 | \( 1 + (-0.666 - 2.48i)T + (-6.06 + 3.5i)T^{2} \) |
| 11 | \( 1 + (-0.332 + 0.255i)T + (2.84 - 10.6i)T^{2} \) |
| 13 | \( 1 + (-3.75 + 4.89i)T + (-3.36 - 12.5i)T^{2} \) |
| 17 | \( 1 - 3.64T + 17T^{2} \) |
| 19 | \( 1 + (-3.97 + 1.64i)T + (13.4 - 13.4i)T^{2} \) |
| 23 | \( 1 + (8.77 + 2.35i)T + (19.9 + 11.5i)T^{2} \) |
| 29 | \( 1 + (-7.27 - 0.957i)T + (28.0 + 7.50i)T^{2} \) |
| 31 | \( 1 + (-1.77 + 1.02i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.36 - 8.12i)T + (-26.1 - 26.1i)T^{2} \) |
| 41 | \( 1 + (-1.45 + 5.42i)T + (-35.5 - 20.5i)T^{2} \) |
| 43 | \( 1 + (2.16 + 2.81i)T + (-11.1 + 41.5i)T^{2} \) |
| 47 | \( 1 + (-2.05 - 1.18i)T + (23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0730 - 0.176i)T + (-37.4 - 37.4i)T^{2} \) |
| 59 | \( 1 + (-5.21 + 0.686i)T + (56.9 - 15.2i)T^{2} \) |
| 61 | \( 1 + (-0.357 + 2.71i)T + (-58.9 - 15.7i)T^{2} \) |
| 67 | \( 1 + (0.969 - 1.26i)T + (-17.3 - 64.7i)T^{2} \) |
| 71 | \( 1 + (-7.18 - 7.18i)T + 71iT^{2} \) |
| 73 | \( 1 + (6.62 - 6.62i)T - 73iT^{2} \) |
| 79 | \( 1 + (4.97 - 8.61i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.26 + 0.561i)T + (80.1 + 21.4i)T^{2} \) |
| 89 | \( 1 + (-2.46 + 2.46i)T - 89iT^{2} \) |
| 97 | \( 1 + (7.74 - 13.4i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.81378259866975623403126413607, −10.71772284438543844627077516321, −9.994629021682848789895702799052, −8.404189948569651819575025218596, −8.200143310170042197885782747227, −6.88130280192745650694612315947, −5.89816671193460010663285383237, −5.28432210700568830233853643160, −2.70156273287821466309970664556, −1.06576512995016296538314701876,
1.25384830723273009938884175589, 3.49771341736972586350394838458, 4.35121662624496115912870939211, 6.02275081133256216856609612149, 7.06376903807357306467171132887, 8.211691284351420455301143174050, 9.281065518026016309411672342487, 10.07414227519767263694632029208, 10.76118819261128848551759036575, 11.66827245011845083207310853345
