| L(s) = 1 | − 9.01·2-s − 31.1·3-s − 46.6·4-s + 514.·5-s + 280.·6-s − 343·7-s + 1.57e3·8-s − 1.21e3·9-s − 4.64e3·10-s + 7.90e3·11-s + 1.45e3·12-s − 2.19e3·13-s + 3.09e3·14-s − 1.60e4·15-s − 8.22e3·16-s − 3.49e4·17-s + 1.09e4·18-s + 4.27e3·19-s − 2.40e4·20-s + 1.06e4·21-s − 7.12e4·22-s − 8.58e4·23-s − 4.90e4·24-s + 1.87e5·25-s + 1.98e4·26-s + 1.06e5·27-s + 1.60e4·28-s + ⋯ |
| L(s) = 1 | − 0.797·2-s − 0.665·3-s − 0.364·4-s + 1.84·5-s + 0.530·6-s − 0.377·7-s + 1.08·8-s − 0.556·9-s − 1.46·10-s + 1.79·11-s + 0.242·12-s − 0.277·13-s + 0.301·14-s − 1.22·15-s − 0.502·16-s − 1.72·17-s + 0.443·18-s + 0.143·19-s − 0.671·20-s + 0.251·21-s − 1.42·22-s − 1.47·23-s − 0.724·24-s + 2.39·25-s + 0.221·26-s + 1.03·27-s + 0.137·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(1.131335383\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.131335383\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + 343T \) |
| 13 | \( 1 + 2.19e3T \) |
| good | 2 | \( 1 + 9.01T + 128T^{2} \) |
| 3 | \( 1 + 31.1T + 2.18e3T^{2} \) |
| 5 | \( 1 - 514.T + 7.81e4T^{2} \) |
| 11 | \( 1 - 7.90e3T + 1.94e7T^{2} \) |
| 17 | \( 1 + 3.49e4T + 4.10e8T^{2} \) |
| 19 | \( 1 - 4.27e3T + 8.93e8T^{2} \) |
| 23 | \( 1 + 8.58e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 1.87e5T + 1.72e10T^{2} \) |
| 31 | \( 1 + 6.73e4T + 2.75e10T^{2} \) |
| 37 | \( 1 - 3.62e5T + 9.49e10T^{2} \) |
| 41 | \( 1 - 2.69e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 5.47e5T + 2.71e11T^{2} \) |
| 47 | \( 1 + 2.19e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 3.30e5T + 1.17e12T^{2} \) |
| 59 | \( 1 - 1.83e6T + 2.48e12T^{2} \) |
| 61 | \( 1 - 5.48e5T + 3.14e12T^{2} \) |
| 67 | \( 1 + 2.38e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 3.21e6T + 9.09e12T^{2} \) |
| 73 | \( 1 - 2.72e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 4.79e6T + 1.92e13T^{2} \) |
| 83 | \( 1 - 3.77e5T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.51e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 3.17e6T + 8.07e13T^{2} \) |
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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
−12.72649449340161646957131582162, −11.34733450555207593288786198371, −10.20258300523258572292990925571, −9.385504974733534495478344043296, −8.728510240469108854133559969085, −6.64617258124150932349738828872, −5.95692632376692712007070619726, −4.48981794086732733184712101048, −2.14147292648031274676478418209, −0.809377055164273670854553376071,
0.809377055164273670854553376071, 2.14147292648031274676478418209, 4.48981794086732733184712101048, 5.95692632376692712007070619726, 6.64617258124150932349738828872, 8.728510240469108854133559969085, 9.385504974733534495478344043296, 10.20258300523258572292990925571, 11.34733450555207593288786198371, 12.72649449340161646957131582162
