| L(s) = 1 | + (−33.8 − 44.4i)5-s + 85.8i·7-s + 488.·11-s − 38.5i·13-s + 692. i·17-s − 2.48e3·19-s − 4.12e3i·23-s + (−830. + 3.01e3i)25-s − 1.88e3·29-s − 2.29e3·31-s + (3.81e3 − 2.90e3i)35-s + 1.06e4i·37-s + 1.52e4·41-s + 9.46e3i·43-s + 1.43e4i·47-s + ⋯ |
| L(s) = 1 | + (−0.605 − 0.795i)5-s + 0.662i·7-s + 1.21·11-s − 0.0633i·13-s + 0.581i·17-s − 1.58·19-s − 1.62i·23-s + (−0.265 + 0.964i)25-s − 0.416·29-s − 0.429·31-s + (0.526 − 0.401i)35-s + 1.27i·37-s + 1.41·41-s + 0.780i·43-s + 0.945i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.795 - 0.605i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.582307470\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.582307470\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (33.8 + 44.4i)T \) |
| good | 7 | \( 1 - 85.8iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 488.T + 1.61e5T^{2} \) |
| 13 | \( 1 + 38.5iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 692. iT - 1.41e6T^{2} \) |
| 19 | \( 1 + 2.48e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 4.12e3iT - 6.43e6T^{2} \) |
| 29 | \( 1 + 1.88e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 2.29e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.06e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 - 1.52e4T + 1.15e8T^{2} \) |
| 43 | \( 1 - 9.46e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 - 1.43e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 1.39e4iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 3.96e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.82e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.41e3iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 1.16e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.10e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 - 4.34e4T + 3.07e9T^{2} \) |
| 83 | \( 1 - 8.66e4iT - 3.93e9T^{2} \) |
| 89 | \( 1 + 1.00e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + 3.99e4iT - 8.58e9T^{2} \) |
| 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
−10.86800755948972199153788655248, −9.615402367229056251377435755519, −8.675928687836215873975190721244, −8.258302016096967383287421567486, −6.80238629942353560790000864776, −5.92997678749114484296412637477, −4.60855020606864027673625494701, −3.86166044105265330315524953504, −2.24363325202322463019501312745, −0.898356521068813594849341273214,
0.52234317014571648782525174908, 2.08392289356223263064983951974, 3.62753032042547365756787678069, 4.18209174092434816452245508904, 5.80388977313753354510710518870, 6.93305739044423728075365321679, 7.42986202262722750502223504425, 8.684797896448448906781313543168, 9.632864933024185725898264264655, 10.70212640420696129953356005635
