| L(s) = 1 | + 2-s + 3-s − 4-s − 2·5-s + 6-s + 2·7-s − 3·8-s + 9-s − 2·10-s − 6·11-s − 12-s + 2·13-s + 2·14-s − 2·15-s − 16-s − 17-s + 18-s − 8·19-s + 2·20-s + 2·21-s − 6·22-s + 6·23-s − 3·24-s − 25-s + 2·26-s + 27-s − 2·28-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 0.577·3-s − 1/2·4-s − 0.894·5-s + 0.408·6-s + 0.755·7-s − 1.06·8-s + 1/3·9-s − 0.632·10-s − 1.80·11-s − 0.288·12-s + 0.554·13-s + 0.534·14-s − 0.516·15-s − 1/4·16-s − 0.242·17-s + 0.235·18-s − 1.83·19-s + 0.447·20-s + 0.436·21-s − 1.27·22-s + 1.25·23-s − 0.612·24-s − 1/5·25-s + 0.392·26-s + 0.192·27-s − 0.377·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 100011 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 - T \) |
| 17 | \( 1 + T \) |
| 37 | \( 1 + T \) |
| 53 | \( 1 - T \) |
| good | 2 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 + 2 T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 2 T + p T^{2} \) |
| 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 | \( 1 + 2 T + p T^{2} \) |
| 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
−13.93802108703619, −13.45596189468759, −13.01015179100566, −12.67278234580962, −12.40504438064274, −11.45180113563216, −11.08493872019481, −10.72974170717674, −10.19737482589787, −9.378046536614405, −8.832141531482335, −8.539671047775760, −8.068876595491768, −7.494788827452983, −7.201777033125233, −6.202442258826556, −5.762856298770622, −5.057787947484691, −4.696233350630486, −4.210936959712628, −3.580972730771657, −3.197309307072032, −2.339331061797025, −1.949115435795926, −0.7068381036555976, 0,
0.7068381036555976, 1.949115435795926, 2.339331061797025, 3.197309307072032, 3.580972730771657, 4.210936959712628, 4.696233350630486, 5.057787947484691, 5.762856298770622, 6.202442258826556, 7.201777033125233, 7.494788827452983, 8.068876595491768, 8.539671047775760, 8.832141531482335, 9.378046536614405, 10.19737482589787, 10.72974170717674, 11.08493872019481, 11.45180113563216, 12.40504438064274, 12.67278234580962, 13.01015179100566, 13.45596189468759, 13.93802108703619
