| L(s) = 1 | − 18·3-s + 160·5-s − 343·7-s − 1.86e3·9-s + 5.70e3·11-s + 1.38e3·13-s − 2.88e3·15-s − 3.14e4·17-s − 1.99e4·19-s + 6.17e3·21-s − 7.71e4·23-s − 5.25e4·25-s + 7.29e4·27-s − 1.93e5·29-s − 2.63e4·31-s − 1.02e5·33-s − 5.48e4·35-s + 2.04e5·37-s − 2.49e4·39-s − 6.63e5·41-s − 3.35e5·43-s − 2.98e5·45-s + 1.11e6·47-s + 1.17e5·49-s + 5.65e5·51-s + 1.12e5·53-s + 9.12e5·55-s + ⋯ |
| L(s) = 1 | − 0.384·3-s + 0.572·5-s − 0.377·7-s − 0.851·9-s + 1.29·11-s + 0.175·13-s − 0.220·15-s − 1.55·17-s − 0.667·19-s + 0.145·21-s − 1.32·23-s − 0.672·25-s + 0.712·27-s − 1.47·29-s − 0.158·31-s − 0.497·33-s − 0.216·35-s + 0.663·37-s − 0.0674·39-s − 1.50·41-s − 0.644·43-s − 0.487·45-s + 1.57·47-s + 1/7·49-s + 0.597·51-s + 0.104·53-s + 0.739·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 56 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 7 | \( 1 + p^{3} T \) |
| good | 3 | \( 1 + 2 p^{2} T + p^{7} T^{2} \) |
| 5 | \( 1 - 32 p T + p^{7} T^{2} \) |
| 11 | \( 1 - 5704 T + p^{7} T^{2} \) |
| 13 | \( 1 - 1388 T + p^{7} T^{2} \) |
| 17 | \( 1 + 31434 T + p^{7} T^{2} \) |
| 19 | \( 1 + 19966 T + p^{7} T^{2} \) |
| 23 | \( 1 + 77136 T + p^{7} T^{2} \) |
| 29 | \( 1 + 193374 T + p^{7} T^{2} \) |
| 31 | \( 1 + 26356 T + p^{7} T^{2} \) |
| 37 | \( 1 - 204346 T + p^{7} T^{2} \) |
| 41 | \( 1 + 663050 T + p^{7} T^{2} \) |
| 43 | \( 1 + 335920 T + p^{7} T^{2} \) |
| 47 | \( 1 - 1119812 T + p^{7} T^{2} \) |
| 53 | \( 1 - 112782 T + p^{7} T^{2} \) |
| 59 | \( 1 - 536154 T + p^{7} T^{2} \) |
| 61 | \( 1 + 1170264 T + p^{7} T^{2} \) |
| 67 | \( 1 - 3890660 T + p^{7} T^{2} \) |
| 71 | \( 1 - 2505344 T + p^{7} T^{2} \) |
| 73 | \( 1 + 1435070 T + p^{7} T^{2} \) |
| 79 | \( 1 - 176536 T + p^{7} T^{2} \) |
| 83 | \( 1 + 6211622 T + p^{7} T^{2} \) |
| 89 | \( 1 + 4729062 T + p^{7} T^{2} \) |
| 97 | \( 1 + 2129562 T + p^{7} T^{2} \) |
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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
−13.29922496171716671467899944084, −11.94445698120192196834600873658, −11.01523449934471614941128848658, −9.603552412424256396136332234511, −8.586044782854710029313847858437, −6.66578607306403642517097803791, −5.79714025906867289637449833236, −3.98609519796225773862196379779, −2.03559170265207450854535141296, 0,
2.03559170265207450854535141296, 3.98609519796225773862196379779, 5.79714025906867289637449833236, 6.66578607306403642517097803791, 8.586044782854710029313847858437, 9.603552412424256396136332234511, 11.01523449934471614941128848658, 11.94445698120192196834600873658, 13.29922496171716671467899944084
