| L(s) = 1 | + (−0.642 + 1.25i)2-s + (1.54 − 0.387i)3-s + (−1.17 − 1.61i)4-s + (−2.15 − 1.01i)5-s + (−0.505 + 2.19i)6-s + (−1.07 − 3.53i)7-s + (2.79 − 0.438i)8-s + (−0.405 + 0.216i)9-s + (2.67 − 2.06i)10-s + (5.71 − 0.847i)11-s + (−2.44 − 2.04i)12-s + (0.924 − 2.58i)13-s + (5.14 + 0.921i)14-s + (−3.72 − 0.741i)15-s + (−1.24 + 3.80i)16-s + (2.46 − 0.489i)17-s + ⋯ |
| L(s) = 1 | + (−0.454 + 0.890i)2-s + (0.892 − 0.223i)3-s + (−0.587 − 0.809i)4-s + (−0.964 − 0.456i)5-s + (−0.206 + 0.896i)6-s + (−0.405 − 1.33i)7-s + (0.987 − 0.155i)8-s + (−0.135 + 0.0723i)9-s + (0.844 − 0.651i)10-s + (1.72 − 0.255i)11-s + (−0.704 − 0.591i)12-s + (0.256 − 0.716i)13-s + (1.37 + 0.246i)14-s + (−0.962 − 0.191i)15-s + (−0.310 + 0.950i)16-s + (0.597 − 0.118i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.821 + 0.570i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.971046 - 0.304263i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.971046 - 0.304263i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.642 - 1.25i)T \) |
| good | 3 | \( 1 + (-1.54 + 0.387i)T + (2.64 - 1.41i)T^{2} \) |
| 5 | \( 1 + (2.15 + 1.01i)T + (3.17 + 3.86i)T^{2} \) |
| 7 | \( 1 + (1.07 + 3.53i)T + (-5.82 + 3.88i)T^{2} \) |
| 11 | \( 1 + (-5.71 + 0.847i)T + (10.5 - 3.19i)T^{2} \) |
| 13 | \( 1 + (-0.924 + 2.58i)T + (-10.0 - 8.24i)T^{2} \) |
| 17 | \( 1 + (-2.46 + 0.489i)T + (15.7 - 6.50i)T^{2} \) |
| 19 | \( 1 + (3.02 + 2.74i)T + (1.86 + 18.9i)T^{2} \) |
| 23 | \( 1 + (1.64 - 0.161i)T + (22.5 - 4.48i)T^{2} \) |
| 29 | \( 1 + (1.72 + 2.32i)T + (-8.41 + 27.7i)T^{2} \) |
| 31 | \( 1 + (-5.42 - 2.24i)T + (21.9 + 21.9i)T^{2} \) |
| 37 | \( 1 + (0.384 - 7.83i)T + (-36.8 - 3.62i)T^{2} \) |
| 41 | \( 1 + (-4.33 + 5.27i)T + (-7.99 - 40.2i)T^{2} \) |
| 43 | \( 1 + (1.99 - 7.98i)T + (-37.9 - 20.2i)T^{2} \) |
| 47 | \( 1 + (-2.91 - 1.94i)T + (17.9 + 43.4i)T^{2} \) |
| 53 | \( 1 + (-4.33 + 5.84i)T + (-15.3 - 50.7i)T^{2} \) |
| 59 | \( 1 + (-4.02 - 11.2i)T + (-45.6 + 37.4i)T^{2} \) |
| 61 | \( 1 + (1.27 - 2.12i)T + (-28.7 - 53.7i)T^{2} \) |
| 67 | \( 1 + (9.33 + 5.59i)T + (31.5 + 59.0i)T^{2} \) |
| 71 | \( 1 + (0.174 - 0.326i)T + (-39.4 - 59.0i)T^{2} \) |
| 73 | \( 1 + (-3.65 + 12.0i)T + (-60.6 - 40.5i)T^{2} \) |
| 79 | \( 1 + (-1.85 - 2.77i)T + (-30.2 + 72.9i)T^{2} \) |
| 83 | \( 1 + (-0.837 - 17.0i)T + (-82.6 + 8.13i)T^{2} \) |
| 89 | \( 1 + (-9.78 - 0.963i)T + (87.2 + 17.3i)T^{2} \) |
| 97 | \( 1 + (0.281 - 0.680i)T + (-68.5 - 68.5i)T^{2} \) |
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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
−11.93147601661619870072054592905, −10.77967032425185617518601456260, −9.687293848516803981036390401951, −8.707101198852492891102891195967, −8.040645256982484648302324107304, −7.23029127515873580837039691530, −6.21925102410718602238155009072, −4.44757070422104439031846399024, −3.58096816329749641531845670020, −0.917795319729001324495592609840,
2.14143338318370027595125210459, 3.45654221061035993039209804696, 4.06771565247582524098422492637, 6.17696505854148817207697709639, 7.56522738059023327946692075429, 8.727034600248686985423265949700, 9.027909633743382877363284159308, 10.01027328361825188460420784911, 11.48275798046506819975067301669, 11.85151355472429586608123063574
