| L(s) = 1 | + 5.90e4·3-s − 3.80e6·5-s − 1.01e9·7-s + 3.48e9·9-s + 1.06e11·11-s + 3.96e11·13-s − 2.24e11·15-s − 3.67e12·17-s − 7.39e12·19-s − 6.01e13·21-s + 2.08e14·23-s − 4.62e14·25-s + 2.05e14·27-s − 1.10e15·29-s − 7.87e15·31-s + 6.28e15·33-s + 3.87e15·35-s + 1.14e16·37-s + 2.34e16·39-s + 9.57e15·41-s − 1.54e17·43-s − 1.32e16·45-s + 2.63e17·47-s + 4.78e17·49-s − 2.17e17·51-s − 1.43e18·53-s − 4.04e17·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 0.174·5-s − 1.36·7-s + 0.333·9-s + 1.23·11-s + 0.798·13-s − 0.100·15-s − 0.442·17-s − 0.276·19-s − 0.786·21-s + 1.04·23-s − 0.969·25-s + 0.192·27-s − 0.487·29-s − 1.72·31-s + 0.713·33-s + 0.237·35-s + 0.392·37-s + 0.461·39-s + 0.111·41-s − 1.09·43-s − 0.0581·45-s + 0.732·47-s + 0.857·49-s − 0.255·51-s − 1.12·53-s − 0.215·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 24 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - 5.90e4T \) |
| good | 5 | \( 1 + 3.80e6T + 4.76e14T^{2} \) |
| 7 | \( 1 + 1.01e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.06e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 3.96e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 3.67e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 7.39e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.08e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 1.10e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 7.87e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 1.14e16T + 8.55e32T^{2} \) |
| 41 | \( 1 - 9.57e15T + 7.38e33T^{2} \) |
| 43 | \( 1 + 1.54e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.63e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.43e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.89e18T + 1.54e37T^{2} \) |
| 61 | \( 1 - 2.96e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 1.75e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 3.35e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 6.12e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 5.83e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 1.25e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.73e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 2.83e20T + 5.27e41T^{2} \) |
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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
−12.64304196186415438913363276923, −11.15931488455148913250714339353, −9.599293077707178620111618979655, −8.814547178336729544070242439236, −7.13594987208896634300936230756, −6.08228467146453273431631027369, −4.03623681697873268399812119612, −3.17012880922289550489294015302, −1.55342846488375356788242113881, 0,
1.55342846488375356788242113881, 3.17012880922289550489294015302, 4.03623681697873268399812119612, 6.08228467146453273431631027369, 7.13594987208896634300936230756, 8.814547178336729544070242439236, 9.599293077707178620111618979655, 11.15931488455148913250714339353, 12.64304196186415438913363276923
