| L(s) = 1 | − 5.09e3·2-s + 1.75e7·4-s + 3.19e7·5-s − 5.17e9·7-s − 4.68e10·8-s − 1.63e11·10-s − 6.04e11·11-s + 7.96e12·13-s + 2.63e13·14-s + 9.13e13·16-s − 1.98e13·17-s + 6.27e14·19-s + 5.62e14·20-s + 3.08e15·22-s + 4.55e15·23-s − 1.08e16·25-s − 4.06e16·26-s − 9.09e16·28-s − 4.14e16·29-s + 1.35e15·31-s − 7.23e16·32-s + 1.01e17·34-s − 1.65e17·35-s + 3.41e17·37-s − 3.19e18·38-s − 1.49e18·40-s + 3.69e18·41-s + ⋯ |
| L(s) = 1 | − 1.75·2-s + 2.09·4-s + 0.293·5-s − 0.988·7-s − 1.92·8-s − 0.515·10-s − 0.638·11-s + 1.23·13-s + 1.73·14-s + 1.29·16-s − 0.140·17-s + 1.23·19-s + 0.614·20-s + 1.12·22-s + 0.997·23-s − 0.914·25-s − 2.16·26-s − 2.07·28-s − 0.630·29-s + 0.00959·31-s − 0.354·32-s + 0.247·34-s − 0.289·35-s + 0.315·37-s − 2.17·38-s − 0.565·40-s + 1.04·41-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(12)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{25}{2})\) |
|
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 | \( 1 + 5.09e3T + 8.38e6T^{2} \) |
| 5 | \( 1 - 3.19e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 5.17e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 6.04e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 7.96e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 1.98e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.27e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 4.55e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 4.14e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.35e15T + 2.00e34T^{2} \) |
| 37 | \( 1 - 3.41e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 3.69e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.96e18T + 3.71e37T^{2} \) |
| 47 | \( 1 + 2.44e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 6.39e19T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.81e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 4.67e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 2.77e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.29e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 4.56e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 3.99e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.45e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.80e21T + 6.85e44T^{2} \) |
| 97 | \( 1 - 8.25e22T + 4.96e45T^{2} \) |
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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
−15.64304371700758468335507662838, −13.25915944396189151729175050636, −11.29870057062710530432850787575, −10.01461704023359497968325649109, −8.989004459822256851516586782378, −7.50776629393555597101016493060, −6.07509836888866812012495849166, −3.01731920984200956466223580015, −1.35294258229057912230533559983, 0,
1.35294258229057912230533559983, 3.01731920984200956466223580015, 6.07509836888866812012495849166, 7.50776629393555597101016493060, 8.989004459822256851516586782378, 10.01461704023359497968325649109, 11.29870057062710530432850787575, 13.25915944396189151729175050636, 15.64304371700758468335507662838
