| L(s) = 1 | − 3-s − 2·9-s + 3·11-s − 6·13-s − 5·17-s − 19-s + 7·23-s + 5·27-s + 2·29-s + 5·31-s − 3·33-s − 3·37-s + 6·39-s + 2·41-s + 4·43-s + 5·47-s + 5·51-s + 53-s + 57-s − 15·59-s + 5·61-s + 9·67-s − 7·69-s + 7·73-s + 79-s + 81-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 2/3·9-s + 0.904·11-s − 1.66·13-s − 1.21·17-s − 0.229·19-s + 1.45·23-s + 0.962·27-s + 0.371·29-s + 0.898·31-s − 0.522·33-s − 0.493·37-s + 0.960·39-s + 0.312·41-s + 0.609·43-s + 0.729·47-s + 0.700·51-s + 0.137·53-s + 0.132·57-s − 1.95·59-s + 0.640·61-s + 1.09·67-s − 0.842·69-s + 0.819·73-s + 0.112·79-s + 1/9·81-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 5 T + p T^{2} \) |
| 19 | \( 1 + T + p T^{2} \) |
| 23 | \( 1 - 7 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 5 T + p T^{2} \) |
| 53 | \( 1 - T + p T^{2} \) |
| 59 | \( 1 + 15 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 - 9 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 7 T + p T^{2} \) |
| 79 | \( 1 - T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 7 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−7.11090715525328506164983029820, −6.67346659900395854996238058076, −6.06939405424016305084870818525, −5.10028871578480742121631367913, −4.78036117456132351094750008146, −3.93847994936428283754598492832, −2.83834001437272116506952471812, −2.32211356741373098005007461439, −1.03451268274920401109328170177, 0,
1.03451268274920401109328170177, 2.32211356741373098005007461439, 2.83834001437272116506952471812, 3.93847994936428283754598492832, 4.78036117456132351094750008146, 5.10028871578480742121631367913, 6.06939405424016305084870818525, 6.67346659900395854996238058076, 7.11090715525328506164983029820
