| L(s) = 1 | − 5-s + 2·7-s − 4·11-s − 8·17-s − 6·19-s + 6·23-s + 25-s + 4·29-s − 2·35-s + 2·37-s − 2·41-s + 4·43-s − 3·49-s + 10·53-s + 4·55-s − 4·59-s − 10·61-s + 12·67-s + 8·71-s + 8·73-s − 8·77-s − 8·79-s − 12·83-s + 8·85-s − 14·89-s + 6·95-s + 16·97-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 1.20·11-s − 1.94·17-s − 1.37·19-s + 1.25·23-s + 1/5·25-s + 0.742·29-s − 0.338·35-s + 0.328·37-s − 0.312·41-s + 0.609·43-s − 3/7·49-s + 1.37·53-s + 0.539·55-s − 0.520·59-s − 1.28·61-s + 1.46·67-s + 0.949·71-s + 0.936·73-s − 0.911·77-s − 0.900·79-s − 1.31·83-s + 0.867·85-s − 1.48·89-s + 0.615·95-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 121680 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 5 | \( 1 + T \) |
| 13 | \( 1 \) |
| good | 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 + 8 T + p T^{2} \) |
| 19 | \( 1 + 6 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 4 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 + 10 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 + 8 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 + 14 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−13.75142027189252, −13.16872440109816, −12.89189098161814, −12.50668936521251, −11.73938709670711, −11.24073589813790, −10.93723974126304, −10.58793281944100, −10.04743099320241, −9.254180528954214, −8.738519437966025, −8.444136101752143, −7.968246945279701, −7.364455009280322, −6.849319124199685, −6.413870043497371, −5.730687941942704, −4.992809641236802, −4.682732624460803, −4.271320444227807, −3.543971866663997, −2.650637417299286, −2.402792693661257, −1.682659352369953, −0.7248150845293749, 0,
0.7248150845293749, 1.682659352369953, 2.402792693661257, 2.650637417299286, 3.543971866663997, 4.271320444227807, 4.682732624460803, 4.992809641236802, 5.730687941942704, 6.413870043497371, 6.849319124199685, 7.364455009280322, 7.968246945279701, 8.444136101752143, 8.738519437966025, 9.254180528954214, 10.04743099320241, 10.58793281944100, 10.93723974126304, 11.24073589813790, 11.73938709670711, 12.50668936521251, 12.89189098161814, 13.16872440109816, 13.75142027189252
