| L(s) = 1 | − 8.68·2-s − 2.07e4·3-s − 1.30e5·4-s − 2.79e5·5-s + 1.80e5·6-s − 8.09e6·7-s + 2.27e6·8-s + 3.02e8·9-s + 2.43e6·10-s + 1.19e8·11-s + 2.72e9·12-s + 4.17e9·13-s + 7.03e7·14-s + 5.81e9·15-s + 1.71e10·16-s − 4.88e10·17-s − 2.62e9·18-s − 1.37e11·19-s + 3.66e10·20-s + 1.68e11·21-s − 1.04e9·22-s − 2.19e10·23-s − 4.72e10·24-s − 6.84e11·25-s − 3.62e10·26-s − 3.59e12·27-s + 1.06e12·28-s + ⋯ |
| L(s) = 1 | − 0.0239·2-s − 1.82·3-s − 0.999·4-s − 0.320·5-s + 0.0438·6-s − 0.530·7-s + 0.0479·8-s + 2.33·9-s + 0.00769·10-s + 0.168·11-s + 1.82·12-s + 1.41·13-s + 0.0127·14-s + 0.585·15-s + 0.998·16-s − 1.69·17-s − 0.0561·18-s − 1.86·19-s + 0.320·20-s + 0.969·21-s − 0.00404·22-s − 0.0585·23-s − 0.0876·24-s − 0.897·25-s − 0.0340·26-s − 2.44·27-s + 0.530·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(18-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 59 ^{s/2} \, \Gamma_{\C}(s+17/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(9)\) |
\(\approx\) |
\(0.006953616853\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.006953616853\) |
| \(L(\frac{19}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 59 | \( 1 - 1.46e14T \) |
| good | 2 | \( 1 + 8.68T + 1.31e5T^{2} \) |
| 3 | \( 1 + 2.07e4T + 1.29e8T^{2} \) |
| 5 | \( 1 + 2.79e5T + 7.62e11T^{2} \) |
| 7 | \( 1 + 8.09e6T + 2.32e14T^{2} \) |
| 11 | \( 1 - 1.19e8T + 5.05e17T^{2} \) |
| 13 | \( 1 - 4.17e9T + 8.65e18T^{2} \) |
| 17 | \( 1 + 4.88e10T + 8.27e20T^{2} \) |
| 19 | \( 1 + 1.37e11T + 5.48e21T^{2} \) |
| 23 | \( 1 + 2.19e10T + 1.41e23T^{2} \) |
| 29 | \( 1 + 2.67e12T + 7.25e24T^{2} \) |
| 31 | \( 1 - 1.33e12T + 2.25e25T^{2} \) |
| 37 | \( 1 - 3.69e13T + 4.56e26T^{2} \) |
| 41 | \( 1 + 7.54e12T + 2.61e27T^{2} \) |
| 43 | \( 1 + 8.66e13T + 5.87e27T^{2} \) |
| 47 | \( 1 + 1.85e14T + 2.66e28T^{2} \) |
| 53 | \( 1 + 6.57e14T + 2.05e29T^{2} \) |
| 61 | \( 1 - 1.29e15T + 2.24e30T^{2} \) |
| 67 | \( 1 + 4.86e15T + 1.10e31T^{2} \) |
| 71 | \( 1 + 4.91e15T + 2.96e31T^{2} \) |
| 73 | \( 1 - 6.78e15T + 4.74e31T^{2} \) |
| 79 | \( 1 - 2.79e15T + 1.81e32T^{2} \) |
| 83 | \( 1 + 3.07e16T + 4.21e32T^{2} \) |
| 89 | \( 1 + 6.14e16T + 1.37e33T^{2} \) |
| 97 | \( 1 - 2.16e16T + 5.95e33T^{2} \) |
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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
−11.47915751753769809377681977132, −10.81370614782748071248050180728, −9.618121692196988584606564753092, −8.338919533501039992077210187299, −6.56733346221668728051290448003, −5.97693572533398914615505611516, −4.57801709322946343634313880615, −3.94339735467239999280512971037, −1.49543805403100334281744396845, −0.04629093360684048474600802996,
0.04629093360684048474600802996, 1.49543805403100334281744396845, 3.94339735467239999280512971037, 4.57801709322946343634313880615, 5.97693572533398914615505611516, 6.56733346221668728051290448003, 8.338919533501039992077210187299, 9.618121692196988584606564753092, 10.81370614782748071248050180728, 11.47915751753769809377681977132
