| L(s) = 1 | − 3-s − 4·7-s + 9-s + 2·11-s + 4·13-s + 4·19-s + 4·21-s − 8·23-s − 27-s − 8·29-s − 2·33-s + 2·37-s − 4·39-s + 4·41-s − 6·43-s + 12·47-s + 9·49-s + 14·53-s − 4·57-s − 2·61-s − 4·63-s − 2·67-s + 8·69-s + 14·71-s − 2·73-s − 8·77-s + 4·79-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.51·7-s + 1/3·9-s + 0.603·11-s + 1.10·13-s + 0.917·19-s + 0.872·21-s − 1.66·23-s − 0.192·27-s − 1.48·29-s − 0.348·33-s + 0.328·37-s − 0.640·39-s + 0.624·41-s − 0.914·43-s + 1.75·47-s + 9/7·49-s + 1.92·53-s − 0.529·57-s − 0.256·61-s − 0.503·63-s − 0.244·67-s + 0.963·69-s + 1.66·71-s − 0.234·73-s − 0.911·77-s + 0.450·79-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 346800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.194068446\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.194068446\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 17 | \( 1 \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 4 T + p T^{2} \) |
| 43 | \( 1 + 6 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 - 14 T + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 14 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
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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
−12.39618897233872, −12.14457257768166, −11.76053233207739, −11.13373178600625, −10.85679715816952, −10.17550129113657, −9.849765812781596, −9.497405349773280, −8.950087272643977, −8.605563064117015, −7.842857217542767, −7.373328283091423, −6.987748439355967, −6.341930826016758, −6.027212711434466, −5.763078436112001, −5.194357161952330, −4.364305470503586, −3.792871657819848, −3.665501784328901, −3.068050861901543, −2.209558775287194, −1.780521700306948, −0.7708990670182339, −0.5675247107531462,
0.5675247107531462, 0.7708990670182339, 1.780521700306948, 2.209558775287194, 3.068050861901543, 3.665501784328901, 3.792871657819848, 4.364305470503586, 5.194357161952330, 5.763078436112001, 6.027212711434466, 6.341930826016758, 6.987748439355967, 7.373328283091423, 7.842857217542767, 8.605563064117015, 8.950087272643977, 9.497405349773280, 9.849765812781596, 10.17550129113657, 10.85679715816952, 11.13373178600625, 11.76053233207739, 12.14457257768166, 12.39618897233872
