| L(s) = 1 | + 3-s + 5-s − 4·7-s − 2·9-s + 2·13-s + 15-s + 2·19-s − 4·21-s + 9·23-s − 4·25-s − 5·27-s − 4·29-s + 5·31-s − 4·35-s − 9·37-s + 2·39-s − 2·41-s + 6·43-s − 2·45-s − 4·47-s + 9·49-s − 6·53-s + 2·57-s − 5·59-s + 8·63-s + 2·65-s − 13·67-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 0.447·5-s − 1.51·7-s − 2/3·9-s + 0.554·13-s + 0.258·15-s + 0.458·19-s − 0.872·21-s + 1.87·23-s − 4/5·25-s − 0.962·27-s − 0.742·29-s + 0.898·31-s − 0.676·35-s − 1.47·37-s + 0.320·39-s − 0.312·41-s + 0.914·43-s − 0.298·45-s − 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.264·57-s − 0.650·59-s + 1.00·63-s + 0.248·65-s − 1.58·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3872 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 - 5 T + p T^{2} \) |
| 37 | \( 1 + 9 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 6 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 + 5 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 + 13 T + p T^{2} \) |
| 71 | \( 1 + T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 10 T + p T^{2} \) |
| 83 | \( 1 + 14 T + p T^{2} \) |
| 89 | \( 1 + 13 T + p T^{2} \) |
| 97 | \( 1 + 19 T + p T^{2} \) |
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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
−8.239008972050577353859246383976, −7.31663710586613146357599820940, −6.63139014326861825379232766476, −5.91605516094976126535681333837, −5.30356031809903737607298868242, −4.05923006715469622798375657472, −3.11217002189154402790703982681, −2.87159521569857720262282496031, −1.51005713766488878934227677949, 0,
1.51005713766488878934227677949, 2.87159521569857720262282496031, 3.11217002189154402790703982681, 4.05923006715469622798375657472, 5.30356031809903737607298868242, 5.91605516094976126535681333837, 6.63139014326861825379232766476, 7.31663710586613146357599820940, 8.239008972050577353859246383976
