| L(s) = 1 | − 3·3-s + 7-s + 6·9-s − 6·13-s + 7·17-s + 5·19-s − 3·21-s − 5·25-s − 9·27-s − 8·29-s + 2·31-s − 4·37-s + 18·39-s + 6·41-s − 3·43-s − 8·47-s + 49-s − 21·51-s − 6·53-s − 15·57-s + 5·59-s − 10·61-s + 6·63-s + 13·67-s + 6·71-s − 11·73-s + 15·75-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s − 1.66·13-s + 1.69·17-s + 1.14·19-s − 0.654·21-s − 25-s − 1.73·27-s − 1.48·29-s + 0.359·31-s − 0.657·37-s + 2.88·39-s + 0.937·41-s − 0.457·43-s − 1.16·47-s + 1/7·49-s − 2.94·51-s − 0.824·53-s − 1.98·57-s + 0.650·59-s − 1.28·61-s + 0.755·63-s + 1.58·67-s + 0.712·71-s − 1.28·73-s + 1.73·75-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 236992 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 23 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 5 T + p T^{2} \) |
| 29 | \( 1 + 8 T + p T^{2} \) |
| 31 | \( 1 - 2 T + p T^{2} \) |
| 37 | \( 1 + 4 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 3 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 5 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 13 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 11 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 + 9 T + p T^{2} \) |
| 89 | \( 1 - 2 T + p T^{2} \) |
| 97 | \( 1 + 18 T + p T^{2} \) |
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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
−12.85909572816665, −12.49260549140695, −12.10414658251709, −11.78612071170602, −11.33247647594408, −11.02884595344245, −10.20381560291619, −10.06992769504583, −9.540402193304327, −9.293980704597889, −8.165912474847002, −7.772531618903229, −7.403869567552806, −7.023172026093663, −6.348965015820317, −5.809366693393119, −5.385726648709417, −5.098142098530606, −4.691549287724100, −3.930327750196070, −3.430200009426834, −2.684380551652875, −1.844598654146048, −1.376470785947651, −0.6359700115464471, 0,
0.6359700115464471, 1.376470785947651, 1.844598654146048, 2.684380551652875, 3.430200009426834, 3.930327750196070, 4.691549287724100, 5.098142098530606, 5.385726648709417, 5.809366693393119, 6.348965015820317, 7.023172026093663, 7.403869567552806, 7.772531618903229, 8.165912474847002, 9.293980704597889, 9.540402193304327, 10.06992769504583, 10.20381560291619, 11.02884595344245, 11.33247647594408, 11.78612071170602, 12.10414658251709, 12.49260549140695, 12.85909572816665
