| L(s) = 1 | + 9.83e5·2-s + 4.17e11·4-s − 1.57e13·5-s + 5.39e15·7-s − 1.30e17·8-s − 1.55e19·10-s + 1.89e20·11-s − 1.17e21·13-s + 5.30e21·14-s − 3.57e23·16-s + 1.39e24·17-s − 2.79e24·19-s − 6.59e24·20-s + 1.86e26·22-s − 5.34e26·23-s − 1.56e27·25-s − 1.15e27·26-s + 2.25e27·28-s − 2.12e28·29-s + 1.34e29·31-s − 2.80e29·32-s + 1.37e30·34-s − 8.51e28·35-s − 4.34e30·37-s − 2.74e30·38-s + 2.05e30·40-s − 3.17e31·41-s + ⋯ |
| L(s) = 1 | + 1.32·2-s + 0.759·4-s − 0.370·5-s + 0.178·7-s − 0.319·8-s − 0.491·10-s + 0.933·11-s − 0.222·13-s + 0.237·14-s − 1.18·16-s + 1.41·17-s − 0.323·19-s − 0.281·20-s + 1.23·22-s − 1.49·23-s − 0.862·25-s − 0.294·26-s + 0.135·28-s − 0.646·29-s + 1.11·31-s − 1.24·32-s + 1.87·34-s − 0.0662·35-s − 1.14·37-s − 0.429·38-s + 0.118·40-s − 1.12·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(40-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+39/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(20)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{41}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| good | 2 | \( 1 - 9.83e5T + 5.49e11T^{2} \) |
| 5 | \( 1 + 1.57e13T + 1.81e27T^{2} \) |
| 7 | \( 1 - 5.39e15T + 9.09e32T^{2} \) |
| 11 | \( 1 - 1.89e20T + 4.11e40T^{2} \) |
| 13 | \( 1 + 1.17e21T + 2.77e43T^{2} \) |
| 17 | \( 1 - 1.39e24T + 9.71e47T^{2} \) |
| 19 | \( 1 + 2.79e24T + 7.43e49T^{2} \) |
| 23 | \( 1 + 5.34e26T + 1.28e53T^{2} \) |
| 29 | \( 1 + 2.12e28T + 1.08e57T^{2} \) |
| 31 | \( 1 - 1.34e29T + 1.45e58T^{2} \) |
| 37 | \( 1 + 4.34e30T + 1.44e61T^{2} \) |
| 41 | \( 1 + 3.17e31T + 7.91e62T^{2} \) |
| 43 | \( 1 + 8.86e31T + 5.07e63T^{2} \) |
| 47 | \( 1 - 3.38e32T + 1.62e65T^{2} \) |
| 53 | \( 1 + 7.04e33T + 1.76e67T^{2} \) |
| 59 | \( 1 + 1.96e34T + 1.15e69T^{2} \) |
| 61 | \( 1 - 6.79e34T + 4.24e69T^{2} \) |
| 67 | \( 1 - 4.10e35T + 1.64e71T^{2} \) |
| 71 | \( 1 + 8.69e35T + 1.58e72T^{2} \) |
| 73 | \( 1 - 3.39e36T + 4.67e72T^{2} \) |
| 79 | \( 1 + 1.47e37T + 1.01e74T^{2} \) |
| 83 | \( 1 + 1.98e37T + 6.98e74T^{2} \) |
| 89 | \( 1 - 1.95e37T + 1.06e76T^{2} \) |
| 97 | \( 1 - 3.46e38T + 3.04e77T^{2} \) |
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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
−12.31111125749915612713495533561, −11.65588016595643027438521711004, −9.804579862624790285268774516852, −8.137496527383678097590627286021, −6.55859582362164202901510335783, −5.40120466672508845341138317873, −4.14942220072611658706091386665, −3.31428431660204359774275833422, −1.72936132001786070772022821403, 0,
1.72936132001786070772022821403, 3.31428431660204359774275833422, 4.14942220072611658706091386665, 5.40120466672508845341138317873, 6.55859582362164202901510335783, 8.137496527383678097590627286021, 9.804579862624790285268774516852, 11.65588016595643027438521711004, 12.31111125749915612713495533561
