| L(s) = 1 | − 3-s + 2·5-s + 9-s + 4·11-s + 2·13-s − 2·15-s − 2·17-s + 4·19-s + 8·23-s − 25-s − 27-s − 6·29-s − 8·31-s − 4·33-s − 2·39-s − 6·41-s − 4·43-s + 2·45-s − 7·49-s + 2·51-s − 2·53-s + 8·55-s − 4·57-s − 4·59-s + 2·61-s + 4·65-s − 4·67-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 0.894·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s − 0.516·15-s − 0.485·17-s + 0.917·19-s + 1.66·23-s − 1/5·25-s − 0.192·27-s − 1.11·29-s − 1.43·31-s − 0.696·33-s − 0.320·39-s − 0.937·41-s − 0.609·43-s + 0.298·45-s − 49-s + 0.280·51-s − 0.274·53-s + 1.07·55-s − 0.529·57-s − 0.520·59-s + 0.256·61-s + 0.496·65-s − 0.488·67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 32856 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 32856 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 37 | \( 1 \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 7 | \( 1 + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 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 | \( 1 + 8 T + p T^{2} \) |
| 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
−15.14875695761782, −14.83166752603851, −14.21047339907985, −13.53235469101025, −13.32392449739639, −12.69353868591232, −12.08255104441318, −11.49151838163745, −11.02029033176656, −10.70861284807158, −9.749634748407465, −9.378541134556226, −9.119656861285300, −8.308766866182437, −7.544117823369096, −6.794039959294607, −6.605434234592384, −5.854882149080785, −5.279782131943014, −4.898777744945370, −3.822263983806840, −3.527287559832197, −2.507057763367590, −1.538872431801983, −1.294194346117160, 0,
1.294194346117160, 1.538872431801983, 2.507057763367590, 3.527287559832197, 3.822263983806840, 4.898777744945370, 5.279782131943014, 5.854882149080785, 6.605434234592384, 6.794039959294607, 7.544117823369096, 8.308766866182437, 9.119656861285300, 9.378541134556226, 9.749634748407465, 10.70861284807158, 11.02029033176656, 11.49151838163745, 12.08255104441318, 12.69353868591232, 13.32392449739639, 13.53235469101025, 14.21047339907985, 14.83166752603851, 15.14875695761782
