| L(s) = 1 | − 8i·3-s + 208i·7-s + 179·9-s + 536·11-s − 694i·13-s − 1.27e3i·17-s + 1.11e3·19-s + 1.66e3·21-s + 3.21e3i·23-s − 3.37e3i·27-s − 2.91e3·29-s + 2.62e3·31-s − 4.28e3i·33-s − 9.45e3i·37-s − 5.55e3·39-s + ⋯ |
| L(s) = 1 | − 0.513i·3-s + 1.60i·7-s + 0.736·9-s + 1.33·11-s − 1.13i·13-s − 1.07i·17-s + 0.706·19-s + 0.823·21-s + 1.26i·23-s − 0.891i·27-s − 0.644·29-s + 0.490·31-s − 0.685i·33-s − 1.13i·37-s − 0.584·39-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 800 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.894 + 0.447i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(2.819438382\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.819438382\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| good | 3 | \( 1 + 8iT - 243T^{2} \) |
| 7 | \( 1 - 208iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 536T + 1.61e5T^{2} \) |
| 13 | \( 1 + 694iT - 3.71e5T^{2} \) |
| 17 | \( 1 + 1.27e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 1.11e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 3.21e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 2.91e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.62e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 9.45e3iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 170T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.99e4iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 32iT - 2.29e8T^{2} \) |
| 53 | \( 1 - 2.21e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 4.14e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 1.54e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.07e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 + 2.85e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 5.36e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 6.91e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.78e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.26e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 6.22e4iT - 8.58e9T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−9.340824335920448743518866328526, −8.736704950128098198739088242230, −7.61120162069170279367033129017, −6.96691888251164231504645833507, −5.81714848316671794806421781432, −5.30280178646016097095698086359, −3.89632223517782213448149176344, −2.80140835190084112013487313543, −1.77481056147175849265754772774, −0.73216932103521270983262236313,
0.917659211573727876752183211192, 1.67900312097751265361853060909, 3.53144618881840690568479977725, 4.15674443373212159223554124100, 4.70968019075414304228255501888, 6.42990435346206177238637998572, 6.81954108633793409855173744391, 7.79984068717065662182975384321, 8.873201340201468672576508167204, 9.768358916214408601910225410688
