| L(s) = 1 | + 106.·2-s + 729·3-s + 3.06e3·4-s + 3.73e4·5-s + 7.73e4·6-s − 4.70e5·7-s − 5.43e5·8-s + 5.31e5·9-s + 3.96e6·10-s − 4.53e6·11-s + 2.23e6·12-s + 2.07e7·13-s − 4.99e7·14-s + 2.72e7·15-s − 8.28e7·16-s + 5.16e7·17-s + 5.63e7·18-s + 2.30e8·19-s + 1.14e8·20-s − 3.43e8·21-s − 4.81e8·22-s − 6.24e7·23-s − 3.96e8·24-s + 1.77e8·25-s + 2.19e9·26-s + 3.87e8·27-s − 1.44e9·28-s + ⋯ |
| L(s) = 1 | + 1.17·2-s + 0.577·3-s + 0.374·4-s + 1.07·5-s + 0.676·6-s − 1.51·7-s − 0.733·8-s + 0.333·9-s + 1.25·10-s − 0.772·11-s + 0.216·12-s + 1.19·13-s − 1.77·14-s + 0.617·15-s − 1.23·16-s + 0.518·17-s + 0.390·18-s + 1.12·19-s + 0.401·20-s − 0.872·21-s − 0.905·22-s − 0.0879·23-s − 0.423·24-s + 0.145·25-s + 1.39·26-s + 0.192·27-s − 0.566·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(14-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3 ^{s/2} \, \Gamma_{\C}(s+13/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(7)\) |
\(\approx\) |
\(2.593234717\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.593234717\) |
| \(L(\frac{15}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 729T \) |
| good | 2 | \( 1 - 106.T + 8.19e3T^{2} \) |
| 5 | \( 1 - 3.73e4T + 1.22e9T^{2} \) |
| 7 | \( 1 + 4.70e5T + 9.68e10T^{2} \) |
| 11 | \( 1 + 4.53e6T + 3.45e13T^{2} \) |
| 13 | \( 1 - 2.07e7T + 3.02e14T^{2} \) |
| 17 | \( 1 - 5.16e7T + 9.90e15T^{2} \) |
| 19 | \( 1 - 2.30e8T + 4.20e16T^{2} \) |
| 23 | \( 1 + 6.24e7T + 5.04e17T^{2} \) |
| 29 | \( 1 + 1.55e9T + 1.02e19T^{2} \) |
| 31 | \( 1 - 1.67e9T + 2.44e19T^{2} \) |
| 37 | \( 1 - 1.13e10T + 2.43e20T^{2} \) |
| 41 | \( 1 - 2.06e10T + 9.25e20T^{2} \) |
| 43 | \( 1 + 4.71e10T + 1.71e21T^{2} \) |
| 47 | \( 1 - 5.74e10T + 5.46e21T^{2} \) |
| 53 | \( 1 + 2.87e11T + 2.60e22T^{2} \) |
| 59 | \( 1 + 5.16e11T + 1.04e23T^{2} \) |
| 61 | \( 1 - 2.96e11T + 1.61e23T^{2} \) |
| 67 | \( 1 - 4.61e11T + 5.48e23T^{2} \) |
| 71 | \( 1 + 1.04e12T + 1.16e24T^{2} \) |
| 73 | \( 1 + 1.68e10T + 1.67e24T^{2} \) |
| 79 | \( 1 + 4.91e11T + 4.66e24T^{2} \) |
| 83 | \( 1 + 2.48e12T + 8.87e24T^{2} \) |
| 89 | \( 1 - 4.61e12T + 2.19e25T^{2} \) |
| 97 | \( 1 + 9.87e12T + 6.73e25T^{2} \) |
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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
−23.27908814589830484320039229010, −21.94549174998010048556732372494, −20.65492061346016610534032817961, −18.45553424075907860268324669499, −15.84343992680003815603690506227, −13.82017621708915447331090651187, −12.96390732240771473294943296203, −9.589044752176290967290333402888, −5.95821493939135373432081779826, −3.16035022213102414206736433855,
3.16035022213102414206736433855, 5.95821493939135373432081779826, 9.589044752176290967290333402888, 12.96390732240771473294943296203, 13.82017621708915447331090651187, 15.84343992680003815603690506227, 18.45553424075907860268324669499, 20.65492061346016610534032817961, 21.94549174998010048556732372494, 23.27908814589830484320039229010
