| L(s) = 1 | + 2.62e5·3-s − 7.64e7·5-s + 4.80e9·7-s − 2.53e10·9-s + 1.05e11·11-s − 1.70e12·13-s − 2.00e13·15-s + 7.28e12·17-s + 4.59e14·19-s + 1.25e15·21-s − 1.23e15·23-s − 6.08e15·25-s − 3.13e16·27-s + 4.27e16·29-s + 1.80e16·31-s + 2.76e16·33-s − 3.66e17·35-s − 1.83e17·37-s − 4.46e17·39-s + 2.74e18·41-s − 1.08e19·43-s + 1.93e18·45-s + 1.91e19·47-s − 4.31e18·49-s + 1.91e18·51-s + 7.00e19·53-s − 8.06e18·55-s + ⋯ |
| L(s) = 1 | + 0.854·3-s − 0.699·5-s + 0.917·7-s − 0.269·9-s + 0.111·11-s − 0.263·13-s − 0.598·15-s + 0.0515·17-s + 0.905·19-s + 0.784·21-s − 0.270·23-s − 0.510·25-s − 1.08·27-s + 0.650·29-s + 0.127·31-s + 0.0953·33-s − 0.642·35-s − 0.169·37-s − 0.225·39-s + 0.777·41-s − 1.77·43-s + 0.188·45-s + 1.13·47-s − 0.157·49-s + 0.0440·51-s + 1.03·53-s − 0.0780·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 2 | \( 1 \) |
| good | 3 | \( 1 - 2.62e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 7.64e7T + 1.19e16T^{2} \) |
| 7 | \( 1 - 4.80e9T + 2.73e19T^{2} \) |
| 11 | \( 1 - 1.05e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 1.70e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 7.28e12T + 1.99e28T^{2} \) |
| 19 | \( 1 - 4.59e14T + 2.57e29T^{2} \) |
| 23 | \( 1 + 1.23e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 4.27e16T + 4.31e33T^{2} \) |
| 31 | \( 1 - 1.80e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.83e17T + 1.17e36T^{2} \) |
| 41 | \( 1 - 2.74e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 1.08e19T + 3.71e37T^{2} \) |
| 47 | \( 1 - 1.91e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 7.00e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.32e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 3.29e18T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.55e20T + 9.99e41T^{2} \) |
| 71 | \( 1 + 3.06e21T + 3.79e42T^{2} \) |
| 73 | \( 1 + 1.76e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 2.27e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 1.79e22T + 1.37e44T^{2} \) |
| 89 | \( 1 - 5.43e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 3.68e22T + 4.96e45T^{2} \) |
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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
−10.02433574189546187597206080414, −8.797729629279178551999156597285, −8.052786264378529134857374549565, −7.26277847194185492262527449895, −5.64392837158175062996980839187, −4.46580475575778370273651288643, −3.44525093672685410565527652376, −2.43372406634686564209997287196, −1.29926803603037278092809935549, 0,
1.29926803603037278092809935549, 2.43372406634686564209997287196, 3.44525093672685410565527652376, 4.46580475575778370273651288643, 5.64392837158175062996980839187, 7.26277847194185492262527449895, 8.052786264378529134857374549565, 8.797729629279178551999156597285, 10.02433574189546187597206080414
