| L(s) = 1 | − 4·7-s − 4·11-s + 13-s + 2·17-s − 4·19-s − 6·29-s − 10·37-s − 2·41-s + 4·43-s + 4·47-s + 9·49-s − 6·53-s + 4·59-s − 2·61-s + 4·67-s + 8·71-s + 2·73-s + 16·77-s + 16·79-s + 4·83-s − 10·89-s − 4·91-s + 2·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 1.51·7-s − 1.20·11-s + 0.277·13-s + 0.485·17-s − 0.917·19-s − 1.11·29-s − 1.64·37-s − 0.312·41-s + 0.609·43-s + 0.583·47-s + 9/7·49-s − 0.824·53-s + 0.520·59-s − 0.256·61-s + 0.488·67-s + 0.949·71-s + 0.234·73-s + 1.82·77-s + 1.80·79-s + 0.439·83-s − 1.05·89-s − 0.419·91-s + 0.203·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 93600 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 13 | \( 1 - T \) |
| good | 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 + 4 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 10 T + p T^{2} \) |
| 41 | \( 1 + 2 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 + 6 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - 2 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 4 T + p T^{2} \) |
| 89 | \( 1 + 10 T + p T^{2} \) |
| 97 | \( 1 - 2 T + p T^{2} \) |
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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
−13.97067688975865, −13.45848564765773, −13.06602029724138, −12.61482424650820, −12.34897753353009, −11.68290306459442, −10.80639063175437, −10.73106867456340, −10.11354178672608, −9.623882000161584, −9.179869053072734, −8.597086979949117, −8.043431947419155, −7.501769381993879, −6.936735685019603, −6.444755805004577, −5.933351870796218, −5.363187350729804, −4.919730423745787, −3.937516012248221, −3.617412750382246, −2.994633107882670, −2.395765505306871, −1.759118778793033, −0.6348922765239105, 0,
0.6348922765239105, 1.759118778793033, 2.395765505306871, 2.994633107882670, 3.617412750382246, 3.937516012248221, 4.919730423745787, 5.363187350729804, 5.933351870796218, 6.444755805004577, 6.936735685019603, 7.501769381993879, 8.043431947419155, 8.597086979949117, 9.179869053072734, 9.623882000161584, 10.11354178672608, 10.73106867456340, 10.80639063175437, 11.68290306459442, 12.34897753353009, 12.61482424650820, 13.06602029724138, 13.45848564765773, 13.97067688975865
