| L(s) = 1 | − 2·5-s − 2·13-s + 2·17-s + 4·19-s − 8·23-s − 25-s − 6·29-s − 8·31-s − 6·37-s − 6·41-s − 4·43-s − 7·49-s − 2·53-s − 4·59-s − 2·61-s + 4·65-s − 4·67-s + 8·71-s − 10·73-s − 8·79-s − 4·83-s − 4·85-s + 6·89-s − 8·95-s + 2·97-s + 101-s + 103-s + ⋯ |
| L(s) = 1 | − 0.894·5-s − 0.554·13-s + 0.485·17-s + 0.917·19-s − 1.66·23-s − 1/5·25-s − 1.11·29-s − 1.43·31-s − 0.986·37-s − 0.937·41-s − 0.609·43-s − 49-s − 0.274·53-s − 0.520·59-s − 0.256·61-s + 0.496·65-s − 0.488·67-s + 0.949·71-s − 1.17·73-s − 0.900·79-s − 0.439·83-s − 0.433·85-s + 0.635·89-s − 0.820·95-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 69696 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 11 | \( 1 \) |
| good | 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−14.64690771482310, −14.25890696598963, −13.62099767265136, −13.19167243370505, −12.46784977546555, −12.09446567142559, −11.72559097173634, −11.25080423022134, −10.65010772206018, −10.01079062336666, −9.670608075849065, −9.073762378434309, −8.418556395620657, −7.863218713472408, −7.528594608141283, −7.082321452447151, −6.340019328068142, −5.665430504777920, −5.229998829286507, −4.598183612764348, −3.712708904471418, −3.660254915278560, −2.845910499425779, −1.910208375633958, −1.444931837568305, 0, 0,
1.444931837568305, 1.910208375633958, 2.845910499425779, 3.660254915278560, 3.712708904471418, 4.598183612764348, 5.229998829286507, 5.665430504777920, 6.340019328068142, 7.082321452447151, 7.528594608141283, 7.863218713472408, 8.418556395620657, 9.073762378434309, 9.670608075849065, 10.01079062336666, 10.65010772206018, 11.25080423022134, 11.72559097173634, 12.09446567142559, 12.46784977546555, 13.19167243370505, 13.62099767265136, 14.25890696598963, 14.64690771482310
