| L(s) = 1 | − 3-s + 5·7-s − 2·9-s + 2·11-s + 3·17-s + 2·19-s − 5·21-s − 4·23-s + 5·27-s − 6·29-s + 4·31-s − 2·33-s + 11·37-s − 8·41-s + 43-s + 9·47-s + 18·49-s − 3·51-s + 12·53-s − 2·57-s − 6·59-s − 10·63-s + 6·67-s + 4·69-s − 7·71-s − 2·73-s + 10·77-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.88·7-s − 2/3·9-s + 0.603·11-s + 0.727·17-s + 0.458·19-s − 1.09·21-s − 0.834·23-s + 0.962·27-s − 1.11·29-s + 0.718·31-s − 0.348·33-s + 1.80·37-s − 1.24·41-s + 0.152·43-s + 1.31·47-s + 18/7·49-s − 0.420·51-s + 1.64·53-s − 0.264·57-s − 0.781·59-s − 1.25·63-s + 0.733·67-s + 0.481·69-s − 0.830·71-s − 0.234·73-s + 1.13·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 33800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.642560101\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.642560101\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 - 5 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 + 8 T + p T^{2} \) |
| 43 | \( 1 - T + p T^{2} \) |
| 47 | \( 1 - 9 T + p T^{2} \) |
| 53 | \( 1 - 12 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 12 T + p T^{2} \) |
| 83 | \( 1 + 16 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 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
−14.96894576670243, −14.48843555365586, −13.92090048163914, −13.71666161637949, −12.77409606963525, −12.02492603982304, −11.72725367270424, −11.49700205048052, −10.81748060546486, −10.38901093092715, −9.658601562813572, −8.978603177525536, −8.462593716009506, −7.878590076171520, −7.544968546908672, −6.748582341712366, −5.936089223786248, −5.568250007972661, −5.062269852241805, −4.335525026416660, −3.870901450497284, −2.855783222804396, −2.113799208349009, −1.347183463394631, −0.6921784560343471,
0.6921784560343471, 1.347183463394631, 2.113799208349009, 2.855783222804396, 3.870901450497284, 4.335525026416660, 5.062269852241805, 5.568250007972661, 5.936089223786248, 6.748582341712366, 7.544968546908672, 7.878590076171520, 8.462593716009506, 8.978603177525536, 9.658601562813572, 10.38901093092715, 10.81748060546486, 11.49700205048052, 11.72725367270424, 12.02492603982304, 12.77409606963525, 13.71666161637949, 13.92090048163914, 14.48843555365586, 14.96894576670243
