| L(s) = 1 | + 3-s − 2·5-s + 4·7-s + 9-s − 4·11-s − 2·15-s − 4·17-s − 4·19-s + 4·21-s + 4·23-s − 25-s + 27-s + 2·29-s − 4·31-s − 4·33-s − 8·35-s + 12·37-s + 12·41-s − 8·43-s − 2·45-s + 9·49-s − 4·51-s − 14·53-s + 8·55-s − 4·57-s + 2·59-s + 2·61-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.894·5-s + 1.51·7-s + 1/3·9-s − 1.20·11-s − 0.516·15-s − 0.970·17-s − 0.917·19-s + 0.872·21-s + 0.834·23-s − 1/5·25-s + 0.192·27-s + 0.371·29-s − 0.718·31-s − 0.696·33-s − 1.35·35-s + 1.97·37-s + 1.87·41-s − 1.21·43-s − 0.298·45-s + 9/7·49-s − 0.560·51-s − 1.92·53-s + 1.07·55-s − 0.529·57-s + 0.260·59-s + 0.256·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32064 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 167 | \( 1 + T \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 13 | \( 1 + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 12 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 + 8 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 2 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 14 T + p T^{2} \) |
| 83 | \( 1 - 6 T + p T^{2} \) |
| 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
−15.15455638703463, −14.89370982006500, −14.43579333999096, −13.77917505160416, −13.09025074569056, −12.86773148602701, −12.15963167540906, −11.36244442461163, −11.09358174603300, −10.78467399404691, −9.992351225555144, −9.219411247374466, −8.718094467428986, −8.110412932883341, −7.788309923916361, −7.465179944708445, −6.588831373369671, −5.888489552069625, −4.951034504195617, −4.649065869955131, −4.139454540996347, −3.316187585824588, −2.469851734177173, −2.060873104949273, −1.051956517441676, 0,
1.051956517441676, 2.060873104949273, 2.469851734177173, 3.316187585824588, 4.139454540996347, 4.649065869955131, 4.951034504195617, 5.888489552069625, 6.588831373369671, 7.465179944708445, 7.788309923916361, 8.110412932883341, 8.718094467428986, 9.219411247374466, 9.992351225555144, 10.78467399404691, 11.09358174603300, 11.36244442461163, 12.15963167540906, 12.86773148602701, 13.09025074569056, 13.77917505160416, 14.43579333999096, 14.89370982006500, 15.15455638703463
