| L(s) = 1 | + 213.·2-s + 6.56e3·3-s − 8.52e4·4-s + 1.40e6·6-s − 7.24e6·7-s − 4.62e7·8-s + 4.30e7·9-s + 4.10e7·11-s − 5.59e8·12-s − 1.35e9·13-s − 1.54e9·14-s + 1.27e9·16-s − 1.84e10·17-s + 9.21e9·18-s + 1.42e9·19-s − 4.75e10·21-s + 8.78e9·22-s + 1.96e10·23-s − 3.03e11·24-s − 2.90e11·26-s + 2.82e11·27-s + 6.17e11·28-s − 3.22e12·29-s − 1.93e12·31-s + 6.34e12·32-s + 2.69e11·33-s − 3.93e12·34-s + ⋯ |
| L(s) = 1 | + 0.590·2-s + 0.577·3-s − 0.650·4-s + 0.341·6-s − 0.474·7-s − 0.975·8-s + 0.333·9-s + 0.0577·11-s − 0.375·12-s − 0.461·13-s − 0.280·14-s + 0.0742·16-s − 0.639·17-s + 0.196·18-s + 0.0192·19-s − 0.274·21-s + 0.0341·22-s + 0.0522·23-s − 0.563·24-s − 0.272·26-s + 0.192·27-s + 0.309·28-s − 1.19·29-s − 0.407·31-s + 1.01·32-s + 0.0333·33-s − 0.378·34-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 75 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(2.099227231\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.099227231\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 - 6.56e3T \) |
| 5 | \( 1 \) |
| good | 2 | \( 1 - 213.T + 1.31e5T^{2} \) |
| 7 | \( 1 + 7.24e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 4.10e7T + 5.05e17T^{2} \) |
| 13 | \( 1 + 1.35e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 1.84e10T + 8.27e20T^{2} \) |
| 19 | \( 1 - 1.42e9T + 5.48e21T^{2} \) |
| 23 | \( 1 - 1.96e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 3.22e12T + 7.25e24T^{2} \) |
| 31 | \( 1 + 1.93e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 2.47e13T + 4.56e26T^{2} \) |
| 41 | \( 1 - 1.91e13T + 2.61e27T^{2} \) |
| 43 | \( 1 - 9.80e12T + 5.87e27T^{2} \) |
| 47 | \( 1 + 6.08e13T + 2.66e28T^{2} \) |
| 53 | \( 1 + 3.72e14T + 2.05e29T^{2} \) |
| 59 | \( 1 - 1.78e15T + 1.27e30T^{2} \) |
| 61 | \( 1 - 1.98e15T + 2.24e30T^{2} \) |
| 67 | \( 1 - 4.22e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 1.65e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 1.26e16T + 4.74e31T^{2} \) |
| 79 | \( 1 + 1.63e16T + 1.81e32T^{2} \) |
| 83 | \( 1 - 2.45e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 4.52e15T + 1.37e33T^{2} \) |
| 97 | \( 1 + 3.59e16T + 5.95e33T^{2} \) |
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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
−11.35748079593489343181932101321, −9.822428804599537207089730540830, −9.143969510982128360877360798199, −8.000701515274392979097313864750, −6.66159079819377007029047681821, −5.40327111521423514234021622593, −4.26659326136886938155860939038, −3.35902959384865326321041139812, −2.21322122286894323423520548478, −0.56636971596168717002925475898,
0.56636971596168717002925475898, 2.21322122286894323423520548478, 3.35902959384865326321041139812, 4.26659326136886938155860939038, 5.40327111521423514234021622593, 6.66159079819377007029047681821, 8.000701515274392979097313864750, 9.143969510982128360877360798199, 9.822428804599537207089730540830, 11.35748079593489343181932101321
