L(s) = 1 | + 2.73i·3-s − i·5-s − 4.73·7-s − 4.46·9-s + 3.46i·11-s + 3.46i·13-s + 2.73·15-s − 3.46·17-s − 2i·19-s − 12.9i·21-s + 2.19·23-s − 25-s − 3.99i·27-s + 2.53·31-s − 9.46·33-s + ⋯ |
L(s) = 1 | + 1.57i·3-s − 0.447i·5-s − 1.78·7-s − 1.48·9-s + 1.04i·11-s + 0.960i·13-s + 0.705·15-s − 0.840·17-s − 0.458i·19-s − 2.82i·21-s + 0.457·23-s − 0.200·25-s − 0.769i·27-s + 0.455·31-s − 1.64·33-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.965 - 0.258i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0968364 + 0.735546i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0968364 + 0.735546i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + iT \) |
good | 3 | \( 1 - 2.73iT - 3T^{2} \) |
| 7 | \( 1 + 4.73T + 7T^{2} \) |
| 11 | \( 1 - 3.46iT - 11T^{2} \) |
| 13 | \( 1 - 3.46iT - 13T^{2} \) |
| 17 | \( 1 + 3.46T + 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 - 2.19T + 23T^{2} \) |
| 29 | \( 1 - 29T^{2} \) |
| 31 | \( 1 - 2.53T + 31T^{2} \) |
| 37 | \( 1 - 6iT - 37T^{2} \) |
| 41 | \( 1 - 9.46T + 41T^{2} \) |
| 43 | \( 1 + 0.196iT - 43T^{2} \) |
| 47 | \( 1 + 2.19T + 47T^{2} \) |
| 53 | \( 1 - 10.3iT - 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 0.928iT - 61T^{2} \) |
| 67 | \( 1 + 0.196iT - 67T^{2} \) |
| 71 | \( 1 + 16.3T + 71T^{2} \) |
| 73 | \( 1 - 6.39T + 73T^{2} \) |
| 79 | \( 1 - 12T + 79T^{2} \) |
| 83 | \( 1 - 1.26iT - 83T^{2} \) |
| 89 | \( 1 + 12.9T + 89T^{2} \) |
| 97 | \( 1 - 14.3T + 97T^{2} \) |
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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
−12.01455838347654272540799682195, −10.88716799671883573125051990121, −9.963652364717962034341806409605, −9.391394021573925929665351354621, −8.885082177889085723789616458715, −7.09376537108382686286186702029, −6.11781683976858902825569331369, −4.72977610001578692730313778408, −4.07204432030604953258822973427, −2.79817035857445385143577081172,
0.50813332105165067251128578781, 2.55409254358087990562653608006, 3.46438869456891020535988927813, 5.81473091863710290136044589843, 6.38324784577413286092374057680, 7.18198777234494538764232766986, 8.168913389320729669025370928443, 9.221406795251264505158662828520, 10.37898493173513427147440889294, 11.33992008615485420208665425748