| L(s) = 1 | + 5-s − 3·7-s + 3·11-s + 13-s − 3·17-s + 4·19-s − 3·23-s + 25-s − 10·29-s − 6·31-s − 3·35-s − 5·37-s + 5·41-s + 2·43-s + 2·47-s + 2·49-s − 11·53-s + 3·55-s + 4·59-s + 61-s + 65-s − 4·67-s + 3·71-s + 6·73-s − 9·77-s + 3·79-s + 16·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s − 1.13·7-s + 0.904·11-s + 0.277·13-s − 0.727·17-s + 0.917·19-s − 0.625·23-s + 1/5·25-s − 1.85·29-s − 1.07·31-s − 0.507·35-s − 0.821·37-s + 0.780·41-s + 0.304·43-s + 0.291·47-s + 2/7·49-s − 1.51·53-s + 0.404·55-s + 0.520·59-s + 0.128·61-s + 0.124·65-s − 0.488·67-s + 0.356·71-s + 0.702·73-s − 1.02·77-s + 0.337·79-s + 1.75·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 3 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 6 T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 - 5 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 2 T + p T^{2} \) |
| 53 | \( 1 + 11 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 3 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 3 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 - 7 T + p T^{2} \) |
| 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
−7.86060902679513953013838807237, −7.10262430838566253068923670003, −6.45983567336832191316179454974, −5.87319596520202643741014741007, −5.11941154229412897009688068511, −3.90280940264400366006159397664, −3.51584305500912206407185489154, −2.39306641677757806079580301562, −1.43254050954790895566573106289, 0,
1.43254050954790895566573106289, 2.39306641677757806079580301562, 3.51584305500912206407185489154, 3.90280940264400366006159397664, 5.11941154229412897009688068511, 5.87319596520202643741014741007, 6.45983567336832191316179454974, 7.10262430838566253068923670003, 7.86060902679513953013838807237
