| L(s) = 1 | + 2·5-s + 7-s − 3·9-s − 11-s − 6·13-s − 2·17-s − 8·19-s − 25-s − 29-s + 4·31-s + 2·35-s + 2·37-s − 2·41-s − 4·43-s − 6·45-s + 4·47-s + 49-s + 10·53-s − 2·55-s + 12·59-s − 2·61-s − 3·63-s − 12·65-s + 12·67-s − 8·71-s − 10·73-s − 77-s + ⋯ |
| L(s) = 1 | + 0.894·5-s + 0.377·7-s − 9-s − 0.301·11-s − 1.66·13-s − 0.485·17-s − 1.83·19-s − 1/5·25-s − 0.185·29-s + 0.718·31-s + 0.338·35-s + 0.328·37-s − 0.312·41-s − 0.609·43-s − 0.894·45-s + 0.583·47-s + 1/7·49-s + 1.37·53-s − 0.269·55-s + 1.56·59-s − 0.256·61-s − 0.377·63-s − 1.48·65-s + 1.46·67-s − 0.949·71-s − 1.17·73-s − 0.113·77-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 142912 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 + T \) |
| 29 | \( 1 + T \) |
| good | 3 | \( 1 + p T^{2} \) |
| 5 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 8 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 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
−13.49455476558962, −13.30205322239467, −12.72447019445404, −12.11761215376756, −11.79207231842395, −11.25924289607009, −10.61930086147478, −10.30487161141777, −9.840984518094288, −9.270089005943632, −8.793784972798622, −8.300903091402721, −7.915712713902716, −7.162948280562493, −6.712534778232092, −6.203836075078255, −5.541051404947979, −5.327685010357727, −4.555300386297429, −4.246236883943064, −3.332035751471061, −2.541667859493332, −2.270756545035979, −1.885911321758809, −0.7025959662799922, 0,
0.7025959662799922, 1.885911321758809, 2.270756545035979, 2.541667859493332, 3.332035751471061, 4.246236883943064, 4.555300386297429, 5.327685010357727, 5.541051404947979, 6.203836075078255, 6.712534778232092, 7.162948280562493, 7.915712713902716, 8.300903091402721, 8.793784972798622, 9.270089005943632, 9.840984518094288, 10.30487161141777, 10.61930086147478, 11.25924289607009, 11.79207231842395, 12.11761215376756, 12.72447019445404, 13.30205322239467, 13.49455476558962
