| L(s) = 1 | + 8.14i·3-s + 11.1·5-s − 63.6i·7-s + 14.7·9-s + 33.3i·11-s − 274.·13-s + 91.0i·15-s − 284.·17-s + 5.17i·19-s + 517.·21-s + 584. i·23-s + 125.·25-s + 779. i·27-s − 344.·29-s + 1.46e3i·31-s + ⋯ |
| L(s) = 1 | + 0.904i·3-s + 0.447·5-s − 1.29i·7-s + 0.181·9-s + 0.275i·11-s − 1.62·13-s + 0.404i·15-s − 0.985·17-s + 0.0143i·19-s + 1.17·21-s + 1.10i·23-s + 0.200·25-s + 1.06i·27-s − 0.409·29-s + 1.52i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 320 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & -\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(0.4987846089\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.4987846089\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 11.1T \) |
| good | 3 | \( 1 - 8.14iT - 81T^{2} \) |
| 7 | \( 1 + 63.6iT - 2.40e3T^{2} \) |
| 11 | \( 1 - 33.3iT - 1.46e4T^{2} \) |
| 13 | \( 1 + 274.T + 2.85e4T^{2} \) |
| 17 | \( 1 + 284.T + 8.35e4T^{2} \) |
| 19 | \( 1 - 5.17iT - 1.30e5T^{2} \) |
| 23 | \( 1 - 584. iT - 2.79e5T^{2} \) |
| 29 | \( 1 + 344.T + 7.07e5T^{2} \) |
| 31 | \( 1 - 1.46e3iT - 9.23e5T^{2} \) |
| 37 | \( 1 + 1.93e3T + 1.87e6T^{2} \) |
| 41 | \( 1 + 976.T + 2.82e6T^{2} \) |
| 43 | \( 1 - 2.04e3iT - 3.41e6T^{2} \) |
| 47 | \( 1 + 2.56e3iT - 4.87e6T^{2} \) |
| 53 | \( 1 + 121.T + 7.89e6T^{2} \) |
| 59 | \( 1 + 3.45e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.13e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + 1.69e3iT - 2.01e7T^{2} \) |
| 71 | \( 1 - 7.64e3iT - 2.54e7T^{2} \) |
| 73 | \( 1 + 6.00e3T + 2.83e7T^{2} \) |
| 79 | \( 1 + 4.01e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + 3.01e3iT - 4.74e7T^{2} \) |
| 89 | \( 1 - 1.19e3T + 6.27e7T^{2} \) |
| 97 | \( 1 + 3.02e3T + 8.85e7T^{2} \) |
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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
−11.18661111960731205951190520171, −10.25280905453538814639561740757, −9.900332558146580586960317933622, −8.923867785851439293764185408919, −7.41201788850195562232524420428, −6.86123012707019493433612027469, −5.15651816088478965052932551912, −4.49468983234525186600576058792, −3.36404346130936773793841523769, −1.69382870318800276925206226611,
0.13673679134951891607173351907, 1.95444514738432696307082485174, 2.59587438902932485331472773582, 4.58954717443043196854091209845, 5.71032966578888816903760465642, 6.63706504806386934139260206981, 7.54762496672813931584903967235, 8.666521451410344597803403568341, 9.462004017865298895625319005979, 10.50564237347018060897092989453
