| L(s) = 1 | − 3-s − 7-s + 9-s − 11-s − 4·13-s − 2·17-s − 4·19-s + 21-s − 2·23-s − 27-s − 2·31-s + 33-s + 8·37-s + 4·39-s − 6·41-s − 4·43-s + 49-s + 2·51-s + 6·53-s + 4·57-s − 4·59-s − 6·61-s − 63-s − 4·67-s + 2·69-s − 4·73-s + 77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.377·7-s + 1/3·9-s − 0.301·11-s − 1.10·13-s − 0.485·17-s − 0.917·19-s + 0.218·21-s − 0.417·23-s − 0.192·27-s − 0.359·31-s + 0.174·33-s + 1.31·37-s + 0.640·39-s − 0.937·41-s − 0.609·43-s + 1/7·49-s + 0.280·51-s + 0.824·53-s + 0.529·57-s − 0.520·59-s − 0.768·61-s − 0.125·63-s − 0.488·67-s + 0.240·69-s − 0.468·73-s + 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 369600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| good | 13 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 6 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 89 | \( 1 + p T^{2} \) |
| 97 | \( 1 + 10 T + p T^{2} \) |
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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
−12.85156917211311, −12.59075948425662, −11.98584132181378, −11.64043273181205, −11.21188169037131, −10.62460567937107, −10.24617184197791, −9.912842684909070, −9.422817549600376, −8.831104200464764, −8.502210916064737, −7.717324382237711, −7.527013795841033, −6.865424553769157, −6.517510670703964, −5.984646967321566, −5.571150598638418, −4.951574973460098, −4.511495837505647, −4.152184604021813, −3.419381189157643, −2.837768209801850, −2.271265296308083, −1.805237703840739, −0.9923970734790979, 0, 0,
0.9923970734790979, 1.805237703840739, 2.271265296308083, 2.837768209801850, 3.419381189157643, 4.152184604021813, 4.511495837505647, 4.951574973460098, 5.571150598638418, 5.984646967321566, 6.517510670703964, 6.865424553769157, 7.527013795841033, 7.717324382237711, 8.502210916064737, 8.831104200464764, 9.422817549600376, 9.912842684909070, 10.24617184197791, 10.62460567937107, 11.21188169037131, 11.64043273181205, 11.98584132181378, 12.59075948425662, 12.85156917211311
