| L(s) = 1 | − 9.01e4·3-s + 8.69e7·5-s − 2.82e9·7-s − 8.60e10·9-s − 1.78e12·11-s + 4.82e12·13-s − 7.83e12·15-s − 7.88e13·17-s + 8.73e14·19-s + 2.54e14·21-s + 6.19e15·23-s − 4.35e15·25-s + 1.62e16·27-s + 1.10e17·29-s + 5.17e16·31-s + 1.61e17·33-s − 2.45e17·35-s − 1.11e18·37-s − 4.34e17·39-s − 2.03e18·41-s + 8.20e17·43-s − 7.48e18·45-s + 2.81e19·47-s − 1.94e19·49-s + 7.10e18·51-s + 4.33e18·53-s − 1.55e20·55-s + ⋯ |
| L(s) = 1 | − 0.293·3-s + 0.796·5-s − 0.539·7-s − 0.913·9-s − 1.89·11-s + 0.746·13-s − 0.233·15-s − 0.557·17-s + 1.71·19-s + 0.158·21-s + 1.35·23-s − 0.365·25-s + 0.561·27-s + 1.68·29-s + 0.365·31-s + 0.555·33-s − 0.429·35-s − 1.03·37-s − 0.219·39-s − 0.576·41-s + 0.134·43-s − 0.727·45-s + 1.65·47-s − 0.709·49-s + 0.163·51-s + 0.0641·53-s − 1.50·55-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 | \( 1 + 9.01e4T + 9.41e10T^{2} \) |
| 5 | \( 1 - 8.69e7T + 1.19e16T^{2} \) |
| 7 | \( 1 + 2.82e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 1.78e12T + 8.95e23T^{2} \) |
| 13 | \( 1 - 4.82e12T + 4.17e25T^{2} \) |
| 17 | \( 1 + 7.88e13T + 1.99e28T^{2} \) |
| 19 | \( 1 - 8.73e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 6.19e15T + 2.08e31T^{2} \) |
| 29 | \( 1 - 1.10e17T + 4.31e33T^{2} \) |
| 31 | \( 1 - 5.17e16T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.11e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 2.03e18T + 1.24e37T^{2} \) |
| 43 | \( 1 - 8.20e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.81e19T + 2.87e38T^{2} \) |
| 53 | \( 1 - 4.33e18T + 4.55e39T^{2} \) |
| 59 | \( 1 + 2.42e20T + 5.36e40T^{2} \) |
| 61 | \( 1 + 1.86e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.37e21T + 9.99e41T^{2} \) |
| 71 | \( 1 - 3.62e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 1.26e21T + 7.18e42T^{2} \) |
| 79 | \( 1 - 2.95e21T + 4.42e43T^{2} \) |
| 83 | \( 1 + 2.85e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 2.88e22T + 6.85e44T^{2} \) |
| 97 | \( 1 + 3.46e22T + 4.96e45T^{2} \) |
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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
−10.23264564060159620910239893314, −9.117121405412684723017320456348, −8.032131280397965921940807823577, −6.70582269136285201942648306568, −5.63094903574349191411298469534, −5.03719602470177028182235377601, −3.15719510674810722177081657681, −2.55349220623605370287637433713, −1.05984496799869358566015390224, 0,
1.05984496799869358566015390224, 2.55349220623605370287637433713, 3.15719510674810722177081657681, 5.03719602470177028182235377601, 5.63094903574349191411298469534, 6.70582269136285201942648306568, 8.032131280397965921940807823577, 9.117121405412684723017320456348, 10.23264564060159620910239893314
