| L(s) = 1 | + (−1 − 2i)5-s + 2i·7-s − 2i·11-s + 2·13-s + 4i·17-s − 2i·19-s − 6i·23-s + (−3 + 4i)25-s − 4i·29-s − 8·31-s + (4 − 2i)35-s + 10·37-s + 6·41-s − 4·43-s − 6i·47-s + ⋯ |
| L(s) = 1 | + (−0.447 − 0.894i)5-s + 0.755i·7-s − 0.603i·11-s + 0.554·13-s + 0.970i·17-s − 0.458i·19-s − 1.25i·23-s + (−0.600 + 0.800i)25-s − 0.742i·29-s − 1.43·31-s + (0.676 − 0.338i)35-s + 1.64·37-s + 0.937·41-s − 0.609·43-s − 0.875i·47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2880 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.316 + 0.948i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.169920302\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.169920302\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1 + 2i)T \) |
| good | 7 | \( 1 - 2iT - 7T^{2} \) |
| 11 | \( 1 + 2iT - 11T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 - 4iT - 17T^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + 6iT - 23T^{2} \) |
| 29 | \( 1 + 4iT - 29T^{2} \) |
| 31 | \( 1 + 8T + 31T^{2} \) |
| 37 | \( 1 - 10T + 37T^{2} \) |
| 41 | \( 1 - 6T + 41T^{2} \) |
| 43 | \( 1 + 4T + 43T^{2} \) |
| 47 | \( 1 + 6iT - 47T^{2} \) |
| 53 | \( 1 + 10T + 53T^{2} \) |
| 59 | \( 1 - 6iT - 59T^{2} \) |
| 61 | \( 1 + 8iT - 61T^{2} \) |
| 67 | \( 1 - 12T + 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 - 73T^{2} \) |
| 79 | \( 1 + 16T + 79T^{2} \) |
| 83 | \( 1 - 4T + 83T^{2} \) |
| 89 | \( 1 + 10T + 89T^{2} \) |
| 97 | \( 1 + 16iT - 97T^{2} \) |
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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
−8.387134035990489276766665418250, −8.190365744772538123403888906739, −7.05396303804935824699683401699, −6.01871320025884937751089199515, −5.61777388354952501180826228999, −4.54022443721985578394127477529, −3.90153461380821647862837790970, −2.81298082593325811837869649026, −1.68190882258874019825918592571, −0.40136974287515611266036619101,
1.22843582981237085624879019230, 2.51531953082232572790009597797, 3.50805525049656096733039729377, 4.08224064862707085530631010825, 5.10412056097083326860946758494, 6.06951324130777776679767519808, 6.88922716980739344009202808527, 7.52000547906358510361120980502, 7.910449631036686579642143372138, 9.162995944366059425466767799339
