| L(s) = 1 | − 3-s − 5-s + 9-s + 6·13-s + 15-s + 2·17-s − 4·19-s − 4·23-s + 25-s − 27-s − 6·29-s − 6·37-s − 6·39-s + 2·41-s − 4·43-s − 45-s + 8·47-s − 2·51-s + 2·53-s + 4·57-s + 12·59-s + 6·61-s − 6·65-s − 4·67-s + 4·69-s + 12·71-s − 10·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s + 1.66·13-s + 0.258·15-s + 0.485·17-s − 0.917·19-s − 0.834·23-s + 1/5·25-s − 0.192·27-s − 1.11·29-s − 0.986·37-s − 0.960·39-s + 0.312·41-s − 0.609·43-s − 0.149·45-s + 1.16·47-s − 0.280·51-s + 0.274·53-s + 0.529·57-s + 1.56·59-s + 0.768·61-s − 0.744·65-s − 0.488·67-s + 0.481·69-s + 1.42·71-s − 1.17·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 47040 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−14.86473954934080, −14.41058532051820, −13.68142401619361, −13.31626195095718, −12.76588726244333, −12.22350563575102, −11.71543755001599, −11.27047825420663, −10.62976011893841, −10.47859114266416, −9.649447500527588, −9.057163719778894, −8.424268432375765, −8.106978958912853, −7.382831390139056, −6.738248061889356, −6.316597932775019, −5.585594049718477, −5.330033656735982, −4.280303275282536, −3.885927339216001, −3.460027254578130, −2.407896997035888, −1.659050927860541, −0.9066396013967288, 0,
0.9066396013967288, 1.659050927860541, 2.407896997035888, 3.460027254578130, 3.885927339216001, 4.280303275282536, 5.330033656735982, 5.585594049718477, 6.316597932775019, 6.738248061889356, 7.382831390139056, 8.106978958912853, 8.424268432375765, 9.057163719778894, 9.649447500527588, 10.47859114266416, 10.62976011893841, 11.27047825420663, 11.71543755001599, 12.22350563575102, 12.76588726244333, 13.31626195095718, 13.68142401619361, 14.41058532051820, 14.86473954934080
