L(s) = 1 | + (1.73 + 1.73i)3-s + (2.44 − 2.44i)5-s − 2.82i·7-s + 2.99i·9-s + (1.73 − 1.73i)11-s + (−2.44 − 2.44i)13-s + 8.48·15-s − 4·17-s + (1.73 + 1.73i)19-s + (4.89 − 4.89i)21-s + 2.82i·23-s − 6.99i·25-s + (−2.44 − 2.44i)29-s + 5.65·31-s + 5.99·33-s + ⋯ |
L(s) = 1 | + (0.999 + 0.999i)3-s + (1.09 − 1.09i)5-s − 1.06i·7-s + 0.999i·9-s + (0.522 − 0.522i)11-s + (−0.679 − 0.679i)13-s + 2.19·15-s − 0.970·17-s + (0.397 + 0.397i)19-s + (1.06 − 1.06i)21-s + 0.589i·23-s − 1.39i·25-s + (−0.454 − 0.454i)29-s + 1.01·31-s + 1.04·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.923 + 0.382i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.639280958\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.639280958\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (-1.73 - 1.73i)T + 3iT^{2} \) |
| 5 | \( 1 + (-2.44 + 2.44i)T - 5iT^{2} \) |
| 7 | \( 1 + 2.82iT - 7T^{2} \) |
| 11 | \( 1 + (-1.73 + 1.73i)T - 11iT^{2} \) |
| 13 | \( 1 + (2.44 + 2.44i)T + 13iT^{2} \) |
| 17 | \( 1 + 4T + 17T^{2} \) |
| 19 | \( 1 + (-1.73 - 1.73i)T + 19iT^{2} \) |
| 23 | \( 1 - 2.82iT - 23T^{2} \) |
| 29 | \( 1 + (2.44 + 2.44i)T + 29iT^{2} \) |
| 31 | \( 1 - 5.65T + 31T^{2} \) |
| 37 | \( 1 + (2.44 - 2.44i)T - 37iT^{2} \) |
| 41 | \( 1 + 2iT - 41T^{2} \) |
| 43 | \( 1 + (8.66 - 8.66i)T - 43iT^{2} \) |
| 47 | \( 1 - 11.3T + 47T^{2} \) |
| 53 | \( 1 + (7.34 - 7.34i)T - 53iT^{2} \) |
| 59 | \( 1 + (-1.73 + 1.73i)T - 59iT^{2} \) |
| 61 | \( 1 + (-2.44 - 2.44i)T + 61iT^{2} \) |
| 67 | \( 1 + (-5.19 - 5.19i)T + 67iT^{2} \) |
| 71 | \( 1 - 2.82iT - 71T^{2} \) |
| 73 | \( 1 - 8iT - 73T^{2} \) |
| 79 | \( 1 + 5.65T + 79T^{2} \) |
| 83 | \( 1 + (-5.19 - 5.19i)T + 83iT^{2} \) |
| 89 | \( 1 - 8iT - 89T^{2} \) |
| 97 | \( 1 - 12T + 97T^{2} \) |
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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
−9.869860552095519648393626501136, −9.153130240550893657988152026354, −8.513105759557639147299275544991, −7.65676896301127112701486781809, −6.41348271840998836025910216288, −5.33685987686578831467231914747, −4.51821725847969150949489188705, −3.73504835690220453964679558553, −2.56678103010906828455253330967, −1.14087809672960643286557244878,
2.01445654326191766998633646123, 2.22050387111419806689243618069, 3.22467301159649448499007806485, 4.84130750712623281423904155249, 6.05038890885419942420865447479, 6.83326223485451916299146043398, 7.19820983513579437119095628655, 8.488847654434064333205621464868, 9.107468367267600352969742203325, 9.746343637998746799808703715955