| L(s) = 1 | − 1.50e4·5-s − 4.80e5·7-s + 7.74e6·11-s + 1.92e6·13-s + 1.58e8·17-s + 2.46e8·19-s − 7.73e8·23-s − 9.95e8·25-s − 2.01e9·29-s + 5.55e9·31-s + 7.21e9·35-s − 1.82e10·37-s + 2.91e10·41-s − 1.47e10·43-s − 1.12e11·47-s + 1.33e11·49-s − 1.09e11·53-s − 1.16e11·55-s + 1.41e11·59-s − 4.17e10·61-s − 2.89e10·65-s + 1.48e11·67-s − 4.72e11·71-s − 8.43e11·73-s − 3.72e12·77-s + 1.72e12·79-s − 4.89e12·83-s + ⋯ |
| L(s) = 1 | − 0.429·5-s − 1.54·7-s + 1.31·11-s + 0.110·13-s + 1.58·17-s + 1.20·19-s − 1.09·23-s − 0.815·25-s − 0.628·29-s + 1.12·31-s + 0.663·35-s − 1.17·37-s + 0.957·41-s − 0.355·43-s − 1.52·47-s + 1.38·49-s − 0.681·53-s − 0.566·55-s + 0.438·59-s − 0.103·61-s − 0.0475·65-s + 0.200·67-s − 0.437·71-s − 0.652·73-s − 2.03·77-s + 0.799·79-s − 1.64·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 72 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + 1.50e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 4.80e5T + 9.68e10T^{2} \) |
| 11 | \( 1 - 7.74e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 1.92e6T + 3.02e14T^{2} \) |
| 17 | \( 1 - 1.58e8T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.46e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 7.73e8T + 5.04e17T^{2} \) |
| 29 | \( 1 + 2.01e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 5.55e9T + 2.44e19T^{2} \) |
| 37 | \( 1 + 1.82e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.91e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 1.47e10T + 1.71e21T^{2} \) |
| 47 | \( 1 + 1.12e11T + 5.46e21T^{2} \) |
| 53 | \( 1 + 1.09e11T + 2.60e22T^{2} \) |
| 59 | \( 1 - 1.41e11T + 1.04e23T^{2} \) |
| 61 | \( 1 + 4.17e10T + 1.61e23T^{2} \) |
| 67 | \( 1 - 1.48e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 4.72e11T + 1.16e24T^{2} \) |
| 73 | \( 1 + 8.43e11T + 1.67e24T^{2} \) |
| 79 | \( 1 - 1.72e12T + 4.66e24T^{2} \) |
| 83 | \( 1 + 4.89e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.21e11T + 2.19e25T^{2} \) |
| 97 | \( 1 - 1.50e13T + 6.73e25T^{2} \) |
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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
−11.71888639398313546994091903726, −9.997587371147553728789162392068, −9.431847976204640915485866209210, −7.919171185735865658900396057091, −6.72090476745379707838741721848, −5.72153617987305969956009192583, −3.88510389884575813793230635352, −3.17002228116505040746618880904, −1.28961485592028742471201154172, 0,
1.28961485592028742471201154172, 3.17002228116505040746618880904, 3.88510389884575813793230635352, 5.72153617987305969956009192583, 6.72090476745379707838741721848, 7.919171185735865658900396057091, 9.431847976204640915485866209210, 9.997587371147553728789162392068, 11.71888639398313546994091903726
