| L(s) = 1 | − 5-s − 3·7-s + 5·11-s + 4·13-s + 3·17-s + 19-s − 8·23-s − 4·25-s − 2·29-s + 4·31-s + 3·35-s − 10·37-s − 10·41-s − 43-s + 47-s + 2·49-s − 4·53-s − 5·55-s + 6·59-s + 13·61-s − 4·65-s + 12·67-s − 2·71-s + 9·73-s − 15·77-s + 8·79-s − 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 1.13·7-s + 1.50·11-s + 1.10·13-s + 0.727·17-s + 0.229·19-s − 1.66·23-s − 4/5·25-s − 0.371·29-s + 0.718·31-s + 0.507·35-s − 1.64·37-s − 1.56·41-s − 0.152·43-s + 0.145·47-s + 2/7·49-s − 0.549·53-s − 0.674·55-s + 0.781·59-s + 1.66·61-s − 0.496·65-s + 1.46·67-s − 0.237·71-s + 1.05·73-s − 1.70·77-s + 0.900·79-s − 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 10944 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 6 T + p T^{2} \) |
| 61 | \( 1 - 13 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−16.66945484699641, −16.19260468105678, −15.70625398812812, −15.28940861298299, −14.37918795491661, −13.89569559302324, −13.53746870071195, −12.62971891061569, −12.17284107468135, −11.66727013323084, −11.18244328748926, −10.12263002540378, −9.883116971668405, −9.245341059972648, −8.365476743555849, −8.155235805642331, −6.955581808767684, −6.692870200318148, −5.977557562731111, −5.394142228928846, −4.175573300131589, −3.642065867288359, −3.381911986353100, −2.039212945877568, −1.166653753785992, 0,
1.166653753785992, 2.039212945877568, 3.381911986353100, 3.642065867288359, 4.175573300131589, 5.394142228928846, 5.977557562731111, 6.692870200318148, 6.955581808767684, 8.155235805642331, 8.365476743555849, 9.245341059972648, 9.883116971668405, 10.12263002540378, 11.18244328748926, 11.66727013323084, 12.17284107468135, 12.62971891061569, 13.53746870071195, 13.89569559302324, 14.37918795491661, 15.28940861298299, 15.70625398812812, 16.19260468105678, 16.66945484699641
