| L(s) = 1 | − 3-s + 2·5-s − 2·7-s + 9-s − 2·15-s + 2·17-s + 6·19-s + 2·21-s + 4·23-s − 25-s − 27-s − 10·29-s − 10·31-s − 4·35-s − 8·37-s + 10·41-s + 4·43-s + 2·45-s − 12·47-s − 3·49-s − 2·51-s − 6·53-s − 6·57-s + 4·59-s + 2·61-s − 2·63-s + 2·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s − 0.755·7-s + 1/3·9-s − 0.516·15-s + 0.485·17-s + 1.37·19-s + 0.436·21-s + 0.834·23-s − 1/5·25-s − 0.192·27-s − 1.85·29-s − 1.79·31-s − 0.676·35-s − 1.31·37-s + 1.56·41-s + 0.609·43-s + 0.298·45-s − 1.75·47-s − 3/7·49-s − 0.280·51-s − 0.824·53-s − 0.794·57-s + 0.520·59-s + 0.256·61-s − 0.251·63-s + 0.244·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8112 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 6 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + 10 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 - 4 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 12 T + p T^{2} \) |
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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
−7.27852076718000459368009391754, −6.82683036718499081362307075956, −5.85539925917255075660818242011, −5.58589908743180347777744581181, −4.93944230955196316631272876743, −3.72492803708044134332727122659, −3.23173944816621120692156216168, −2.09102081643941051798823853933, −1.27835438490651664139284797652, 0,
1.27835438490651664139284797652, 2.09102081643941051798823853933, 3.23173944816621120692156216168, 3.72492803708044134332727122659, 4.93944230955196316631272876743, 5.58589908743180347777744581181, 5.85539925917255075660818242011, 6.82683036718499081362307075956, 7.27852076718000459368009391754
