| L(s) = 1 | − 3-s − 5-s + 9-s − 2·11-s − 2·13-s + 15-s + 4·17-s − 8·23-s + 25-s − 27-s − 2·31-s + 2·33-s + 8·37-s + 2·39-s + 2·41-s + 2·43-s − 45-s + 10·47-s − 4·51-s − 2·53-s + 2·55-s + 4·59-s + 10·61-s + 2·65-s − 2·67-s + 8·69-s + 12·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 0.447·5-s + 1/3·9-s − 0.603·11-s − 0.554·13-s + 0.258·15-s + 0.970·17-s − 1.66·23-s + 1/5·25-s − 0.192·27-s − 0.359·31-s + 0.348·33-s + 1.31·37-s + 0.320·39-s + 0.312·41-s + 0.304·43-s − 0.149·45-s + 1.45·47-s − 0.560·51-s − 0.274·53-s + 0.269·55-s + 0.520·59-s + 1.28·61-s + 0.248·65-s − 0.244·67-s + 0.963·69-s + 1.42·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 11760 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + T \) |
| 5 | \( 1 + T \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + 8 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 37 | \( 1 - 8 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 2 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 2 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 + 10 T + p T^{2} \) |
| 79 | \( 1 + 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 + 6 T + p T^{2} \) |
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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
−16.55298991102113, −16.15174899248767, −15.73656563439897, −14.97898124839541, −14.48651889176804, −13.90126313826654, −13.15814627166376, −12.53842933641860, −12.13804131405989, −11.58033225389839, −10.96397362534330, −10.29828442599983, −9.861400248182746, −9.222649688053764, −8.233664831197543, −7.826857294237228, −7.275318624464246, −6.512339263606177, −5.653834993367279, −5.402896352142858, −4.348095488619074, −3.950784195643372, −2.908520724513745, −2.150615578625275, −0.9930103499931771, 0,
0.9930103499931771, 2.150615578625275, 2.908520724513745, 3.950784195643372, 4.348095488619074, 5.402896352142858, 5.653834993367279, 6.512339263606177, 7.275318624464246, 7.826857294237228, 8.233664831197543, 9.222649688053764, 9.861400248182746, 10.29828442599983, 10.96397362534330, 11.58033225389839, 12.13804131405989, 12.53842933641860, 13.15814627166376, 13.90126313826654, 14.48651889176804, 14.97898124839541, 15.73656563439897, 16.15174899248767, 16.55298991102113
