| L(s) = 1 | − 3·3-s + 7-s + 6·9-s − 5·11-s − 4·17-s + 2·19-s − 3·21-s − 5·23-s − 5·25-s − 9·27-s + 4·29-s + 31-s + 15·33-s − 7·37-s + 9·41-s + 12·43-s − 7·47-s + 49-s + 12·51-s − 4·53-s − 6·57-s − 6·59-s + 13·61-s + 6·63-s + 11·67-s + 15·69-s − 7·73-s + ⋯ |
| L(s) = 1 | − 1.73·3-s + 0.377·7-s + 2·9-s − 1.50·11-s − 0.970·17-s + 0.458·19-s − 0.654·21-s − 1.04·23-s − 25-s − 1.73·27-s + 0.742·29-s + 0.179·31-s + 2.61·33-s − 1.15·37-s + 1.40·41-s + 1.82·43-s − 1.02·47-s + 1/7·49-s + 1.68·51-s − 0.549·53-s − 0.794·57-s − 0.781·59-s + 1.66·61-s + 0.755·63-s + 1.34·67-s + 1.80·69-s − 0.819·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 18928 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 - T \) |
| 13 | \( 1 \) |
| good | 3 | \( 1 + p T + p T^{2} \) |
| 5 | \( 1 + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 + 4 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 9 T + p T^{2} \) |
| 43 | \( 1 - 12 T + p T^{2} \) |
| 47 | \( 1 + 7 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 + 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 11 T + p T^{2} \) |
| 71 | \( 1 + p T^{2} \) |
| 73 | \( 1 + 7 T + p T^{2} \) |
| 79 | \( 1 - 17 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 5 T + p T^{2} \) |
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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
−16.06642294383324, −15.59621176438472, −15.35544385507410, −14.09404512990058, −13.93254025174538, −12.93435281011904, −12.73528500194312, −12.08516741011318, −11.46794040900017, −11.11487419737247, −10.55224730454876, −10.10531622839055, −9.538131935118055, −8.613791401563375, −7.834710731580024, −7.502804089093432, −6.654588678819167, −6.118612573548972, −5.540045760989984, −5.048565798983999, −4.481857898225361, −3.799010382066435, −2.582939121150745, −1.899409397695205, −0.7851500056035390, 0,
0.7851500056035390, 1.899409397695205, 2.582939121150745, 3.799010382066435, 4.481857898225361, 5.048565798983999, 5.540045760989984, 6.118612573548972, 6.654588678819167, 7.502804089093432, 7.834710731580024, 8.613791401563375, 9.538131935118055, 10.10531622839055, 10.55224730454876, 11.11487419737247, 11.46794040900017, 12.08516741011318, 12.73528500194312, 12.93435281011904, 13.93254025174538, 14.09404512990058, 15.35544385507410, 15.59621176438472, 16.06642294383324
