| L(s) = 1 | − 2.16e5·3-s − 1.90e8·5-s + 9.74e9·7-s − 4.70e10·9-s − 9.58e11·11-s + 2.14e12·13-s + 4.14e13·15-s + 2.65e13·17-s − 7.63e14·19-s − 2.11e15·21-s + 3.75e15·23-s + 2.45e16·25-s + 3.06e16·27-s − 5.01e16·29-s − 1.90e17·31-s + 2.07e17·33-s − 1.86e18·35-s + 1.56e18·37-s − 4.65e17·39-s − 3.05e18·41-s − 3.07e18·43-s + 8.98e18·45-s + 2.72e18·47-s + 6.75e19·49-s − 5.76e18·51-s + 2.52e19·53-s + 1.83e20·55-s + ⋯ |
| L(s) = 1 | − 0.707·3-s − 1.74·5-s + 1.86·7-s − 0.499·9-s − 1.01·11-s + 0.331·13-s + 1.23·15-s + 0.187·17-s − 1.50·19-s − 1.31·21-s + 0.822·23-s + 2.05·25-s + 1.06·27-s − 0.762·29-s − 1.34·31-s + 0.716·33-s − 3.25·35-s + 1.44·37-s − 0.234·39-s − 0.867·41-s − 0.504·43-s + 0.874·45-s + 0.160·47-s + 2.46·49-s − 0.132·51-s + 0.374·53-s + 1.77·55-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 | \( 1 + 2.16e5T + 9.41e10T^{2} \) |
| 5 | \( 1 + 1.90e8T + 1.19e16T^{2} \) |
| 7 | \( 1 - 9.74e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 9.58e11T + 8.95e23T^{2} \) |
| 13 | \( 1 - 2.14e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.65e13T + 1.99e28T^{2} \) |
| 19 | \( 1 + 7.63e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 3.75e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 5.01e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.90e17T + 2.00e34T^{2} \) |
| 37 | \( 1 - 1.56e18T + 1.17e36T^{2} \) |
| 41 | \( 1 + 3.05e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 3.07e18T + 3.71e37T^{2} \) |
| 47 | \( 1 - 2.72e18T + 2.87e38T^{2} \) |
| 53 | \( 1 - 2.52e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 2.29e20T + 5.36e40T^{2} \) |
| 61 | \( 1 - 6.13e20T + 1.15e41T^{2} \) |
| 67 | \( 1 - 1.20e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 2.43e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.45e20T + 7.18e42T^{2} \) |
| 79 | \( 1 - 3.08e21T + 4.42e43T^{2} \) |
| 83 | \( 1 - 1.36e22T + 1.37e44T^{2} \) |
| 89 | \( 1 + 1.67e22T + 6.85e44T^{2} \) |
| 97 | \( 1 - 7.71e22T + 4.96e45T^{2} \) |
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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
−10.89217414920002178811394964191, −8.515776796078543623802889512660, −8.106485938075493174392233539827, −7.10090319325930114759093630806, −5.45357764397238447922764264822, −4.72598490814496730204535787724, −3.75874290092946842197733443766, −2.28175877978117703984477179793, −0.868951464312447594541301041790, 0,
0.868951464312447594541301041790, 2.28175877978117703984477179793, 3.75874290092946842197733443766, 4.72598490814496730204535787724, 5.45357764397238447922764264822, 7.10090319325930114759093630806, 8.106485938075493174392233539827, 8.515776796078543623802889512660, 10.89217414920002178811394964191
