| L(s) = 1 | − 9.55e15·2-s − 1.37e25·3-s + 5.07e31·4-s − 2.80e36·5-s + 1.31e41·6-s − 2.78e44·7-s − 9.70e46·8-s + 6.48e49·9-s + 2.67e52·10-s + 6.08e53·11-s − 6.99e56·12-s − 5.49e58·13-s + 2.65e60·14-s + 3.86e61·15-s − 1.13e63·16-s − 2.90e64·17-s − 6.19e65·18-s − 1.56e67·19-s − 1.42e68·20-s + 3.83e69·21-s − 5.81e69·22-s + 5.72e71·23-s + 1.33e72·24-s − 1.68e73·25-s + 5.24e74·26-s + 8.33e74·27-s − 1.41e76·28-s + ⋯ |
| L(s) = 1 | − 1.50·2-s − 1.23·3-s + 1.25·4-s − 0.564·5-s + 1.84·6-s − 1.19·7-s − 0.375·8-s + 0.517·9-s + 0.846·10-s + 0.129·11-s − 1.54·12-s − 1.80·13-s + 1.78·14-s + 0.694·15-s − 0.686·16-s − 0.731·17-s − 0.776·18-s − 1.14·19-s − 0.705·20-s + 1.46·21-s − 0.193·22-s + 1.84·23-s + 0.462·24-s − 0.681·25-s + 2.71·26-s + 0.594·27-s − 1.49·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s) \, L(s)\cr=\mathstrut & -\,\Lambda(106-s)\end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut &\Gamma_{\C}(s+52.5) \, L(s)\cr=\mathstrut & -\,\Lambda(1-s)\end{aligned}\]
Particular Values
| \(L(53)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{107}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| good | 2 | \( 1 + 9.55e15T + 4.05e31T^{2} \) |
| 3 | \( 1 + 1.37e25T + 1.25e50T^{2} \) |
| 5 | \( 1 + 2.80e36T + 2.46e73T^{2} \) |
| 7 | \( 1 + 2.78e44T + 5.43e88T^{2} \) |
| 11 | \( 1 - 6.08e53T + 2.21e109T^{2} \) |
| 13 | \( 1 + 5.49e58T + 9.20e116T^{2} \) |
| 17 | \( 1 + 2.90e64T + 1.57e129T^{2} \) |
| 19 | \( 1 + 1.56e67T + 1.85e134T^{2} \) |
| 23 | \( 1 - 5.72e71T + 9.58e142T^{2} \) |
| 29 | \( 1 + 1.76e76T + 3.56e153T^{2} \) |
| 31 | \( 1 - 1.61e78T + 3.91e156T^{2} \) |
| 37 | \( 1 - 1.96e82T + 4.58e164T^{2} \) |
| 41 | \( 1 - 1.17e83T + 2.19e169T^{2} \) |
| 43 | \( 1 - 1.98e84T + 3.26e171T^{2} \) |
| 47 | \( 1 - 1.10e88T + 3.71e175T^{2} \) |
| 53 | \( 1 + 7.34e89T + 1.11e181T^{2} \) |
| 59 | \( 1 + 1.37e93T + 8.69e185T^{2} \) |
| 61 | \( 1 - 1.74e93T + 2.88e187T^{2} \) |
| 67 | \( 1 - 1.25e96T + 5.46e191T^{2} \) |
| 71 | \( 1 + 1.81e97T + 2.41e194T^{2} \) |
| 73 | \( 1 + 2.37e97T + 4.45e195T^{2} \) |
| 79 | \( 1 - 8.68e98T + 1.78e199T^{2} \) |
| 83 | \( 1 - 2.69e100T + 3.18e201T^{2} \) |
| 89 | \( 1 + 5.78e101T + 4.85e204T^{2} \) |
| 97 | \( 1 - 1.53e103T + 4.08e208T^{2} \) |
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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
−12.66323796617456848954104266175, −11.37408163513349698196670288558, −10.24190745145928612759084338581, −9.083051712698212924923901800579, −7.36176336770378457356853210865, −6.41211190653736602546487150668, −4.62962452377337361497565598717, −2.52156215037672634036692424523, −0.65376610677889205136415830235, 0,
0.65376610677889205136415830235, 2.52156215037672634036692424523, 4.62962452377337361497565598717, 6.41211190653736602546487150668, 7.36176336770378457356853210865, 9.083051712698212924923901800579, 10.24190745145928612759084338581, 11.37408163513349698196670288558, 12.66323796617456848954104266175
