| L(s) = 1 | − 3.88e5·3-s + 1.05e8·5-s − 3.81e9·7-s + 5.67e10·9-s − 2.52e11·11-s − 3.59e12·13-s − 4.08e13·15-s + 2.34e14·17-s + 6.23e14·19-s + 1.48e15·21-s + 3.58e15·23-s − 8.82e14·25-s + 1.45e16·27-s − 2.05e16·29-s − 1.36e17·31-s + 9.79e16·33-s − 4.00e17·35-s − 1.23e18·37-s + 1.39e18·39-s + 1.40e18·41-s − 2.18e17·43-s + 5.96e18·45-s + 8.67e18·47-s − 1.28e19·49-s − 9.09e19·51-s − 7.63e19·53-s − 2.64e19·55-s + ⋯ |
| L(s) = 1 | − 1.26·3-s + 0.962·5-s − 0.728·7-s + 0.602·9-s − 0.266·11-s − 0.555·13-s − 1.21·15-s + 1.65·17-s + 1.22·19-s + 0.922·21-s + 0.785·23-s − 0.0740·25-s + 0.502·27-s − 0.313·29-s − 0.963·31-s + 0.337·33-s − 0.701·35-s − 1.14·37-s + 0.703·39-s + 0.397·41-s − 0.0359·43-s + 0.580·45-s + 0.512·47-s − 0.469·49-s − 2.09·51-s − 1.13·53-s − 0.256·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(24-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 16 ^{s/2} \, \Gamma_{\C}(s+23/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 3.88e5T + 9.41e10T^{2} \) |
| 5 | \( 1 - 1.05e8T + 1.19e16T^{2} \) |
| 7 | \( 1 + 3.81e9T + 2.73e19T^{2} \) |
| 11 | \( 1 + 2.52e11T + 8.95e23T^{2} \) |
| 13 | \( 1 + 3.59e12T + 4.17e25T^{2} \) |
| 17 | \( 1 - 2.34e14T + 1.99e28T^{2} \) |
| 19 | \( 1 - 6.23e14T + 2.57e29T^{2} \) |
| 23 | \( 1 - 3.58e15T + 2.08e31T^{2} \) |
| 29 | \( 1 + 2.05e16T + 4.31e33T^{2} \) |
| 31 | \( 1 + 1.36e17T + 2.00e34T^{2} \) |
| 37 | \( 1 + 1.23e18T + 1.17e36T^{2} \) |
| 41 | \( 1 - 1.40e18T + 1.24e37T^{2} \) |
| 43 | \( 1 + 2.18e17T + 3.71e37T^{2} \) |
| 47 | \( 1 - 8.67e18T + 2.87e38T^{2} \) |
| 53 | \( 1 + 7.63e19T + 4.55e39T^{2} \) |
| 59 | \( 1 - 1.01e18T + 5.36e40T^{2} \) |
| 61 | \( 1 - 2.87e20T + 1.15e41T^{2} \) |
| 67 | \( 1 + 1.47e21T + 9.99e41T^{2} \) |
| 71 | \( 1 + 7.64e20T + 3.79e42T^{2} \) |
| 73 | \( 1 + 3.49e21T + 7.18e42T^{2} \) |
| 79 | \( 1 + 1.02e22T + 4.42e43T^{2} \) |
| 83 | \( 1 - 7.71e21T + 1.37e44T^{2} \) |
| 89 | \( 1 - 4.58e21T + 6.85e44T^{2} \) |
| 97 | \( 1 + 1.13e23T + 4.96e45T^{2} \) |
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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
−12.90706692586499688421606331728, −11.84210463950351238458279361050, −10.39819033798327568298290185448, −9.487130186096180434978851802582, −7.26316909152946154236682831623, −5.87434013708782788777677155496, −5.22621896371540424009009089828, −3.09980718338561759622781009122, −1.31313979235274195425059904199, 0,
1.31313979235274195425059904199, 3.09980718338561759622781009122, 5.22621896371540424009009089828, 5.87434013708782788777677155496, 7.26316909152946154236682831623, 9.487130186096180434978851802582, 10.39819033798327568298290185448, 11.84210463950351238458279361050, 12.90706692586499688421606331728
