| L(s) = 1 | − 5.90e4·3-s + 4.18e7·5-s − 6.87e8·7-s + 3.48e9·9-s + 1.49e9·11-s − 8.84e9·13-s − 2.46e12·15-s − 9.41e12·17-s − 4.23e13·19-s + 4.05e13·21-s + 1.09e14·23-s + 1.27e15·25-s − 2.05e14·27-s + 1.18e15·29-s − 1.95e15·31-s − 8.84e13·33-s − 2.87e16·35-s + 4.20e16·37-s + 5.22e14·39-s − 6.92e16·41-s + 8.95e16·43-s + 1.45e17·45-s − 5.26e17·47-s − 8.63e16·49-s + 5.55e17·51-s − 2.24e18·53-s + 6.26e16·55-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.91·5-s − 0.919·7-s + 0.333·9-s + 0.0174·11-s − 0.0178·13-s − 1.10·15-s − 1.13·17-s − 1.58·19-s + 0.530·21-s + 0.550·23-s + 2.66·25-s − 0.192·27-s + 0.521·29-s − 0.427·31-s − 0.0100·33-s − 1.76·35-s + 1.43·37-s + 0.0102·39-s − 0.805·41-s + 0.631·43-s + 0.638·45-s − 1.46·47-s − 0.154·49-s + 0.653·51-s − 1.76·53-s + 0.0333·55-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 | \( 1 - 4.18e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 6.87e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.49e9T + 7.40e21T^{2} \) |
| 13 | \( 1 + 8.84e9T + 2.47e23T^{2} \) |
| 17 | \( 1 + 9.41e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 4.23e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 1.09e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.18e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 1.95e15T + 2.08e31T^{2} \) |
| 37 | \( 1 - 4.20e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 6.92e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 8.95e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 5.26e17T + 1.30e35T^{2} \) |
| 53 | \( 1 + 2.24e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.41e17T + 1.54e37T^{2} \) |
| 61 | \( 1 - 3.17e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 2.04e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 3.92e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 4.75e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 8.63e18T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.65e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 4.57e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 9.21e20T + 5.27e41T^{2} \) |
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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
−12.83079365200950458693596220991, −10.90441201662208846479299128515, −9.924014153411425710143387184409, −8.958808748928458335232701807835, −6.60182002609574735257649090827, −6.10509445143669605602281602265, −4.70466507627603774608404608474, −2.70027511312130115226258086909, −1.57567220201664377605531333671, 0,
1.57567220201664377605531333671, 2.70027511312130115226258086909, 4.70466507627603774608404608474, 6.10509445143669605602281602265, 6.60182002609574735257649090827, 8.958808748928458335232701807835, 9.924014153411425710143387184409, 10.90441201662208846479299128515, 12.83079365200950458693596220991
