| L(s) = 1 | + 8.43i·3-s − 12.2i·5-s − 1.63·7-s − 44.1·9-s + 25.7i·11-s − 13.2i·13-s + 103.·15-s − 53.6·17-s + 100. i·19-s − 13.8i·21-s − 25.1·23-s − 25.6·25-s − 144. i·27-s + 256. i·29-s + 132.·31-s + ⋯ |
| L(s) = 1 | + 1.62i·3-s − 1.09i·5-s − 0.0885·7-s − 1.63·9-s + 0.705i·11-s − 0.282i·13-s + 1.78·15-s − 0.764·17-s + 1.21i·19-s − 0.143i·21-s − 0.227·23-s − 0.205·25-s − 1.03i·27-s + 1.63i·29-s + 0.768·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1024 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & i\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.3126089096\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.3126089096\) |
| \(L(\frac{5}{2})\) |
|
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 | \( 1 - 8.43iT - 27T^{2} \) |
| 5 | \( 1 + 12.2iT - 125T^{2} \) |
| 7 | \( 1 + 1.63T + 343T^{2} \) |
| 11 | \( 1 - 25.7iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 13.2iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 53.6T + 4.91e3T^{2} \) |
| 19 | \( 1 - 100. iT - 6.85e3T^{2} \) |
| 23 | \( 1 + 25.1T + 1.21e4T^{2} \) |
| 29 | \( 1 - 256. iT - 2.43e4T^{2} \) |
| 31 | \( 1 - 132.T + 2.97e4T^{2} \) |
| 37 | \( 1 + 247. iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 198.T + 6.89e4T^{2} \) |
| 43 | \( 1 + 404. iT - 7.95e4T^{2} \) |
| 47 | \( 1 + 78.3T + 1.03e5T^{2} \) |
| 53 | \( 1 + 743. iT - 1.48e5T^{2} \) |
| 59 | \( 1 + 65.8iT - 2.05e5T^{2} \) |
| 61 | \( 1 + 273. iT - 2.26e5T^{2} \) |
| 67 | \( 1 + 399. iT - 3.00e5T^{2} \) |
| 71 | \( 1 + 727.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 106.T + 3.89e5T^{2} \) |
| 79 | \( 1 - 58.9T + 4.93e5T^{2} \) |
| 83 | \( 1 + 580. iT - 5.71e5T^{2} \) |
| 89 | \( 1 - 768.T + 7.04e5T^{2} \) |
| 97 | \( 1 + 809.T + 9.12e5T^{2} \) |
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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
−9.354132879851938128649215074098, −8.789421873502167065082645204738, −8.042844618980705963693605373968, −6.71749437992052652466180307560, −5.44980105650752861835068729729, −4.94826443983000665887488946894, −4.15286644242156118598265950375, −3.33312006195568292232155480244, −1.77116347144717109233671762589, −0.080025516966570169121561703355,
1.15627354942952423732736285470, 2.45927271131856867061771147925, 2.97919269177880526235097329091, 4.51848012279439401442769510107, 6.03586488121605601362038320102, 6.48409582046857299660535975975, 7.11640913300218700285076783699, 7.946786840375326299185446266064, 8.670565612475386667165874002563, 9.766403392110889442208503479221
