| L(s) = 1 | + 5.90e4·3-s + 2.53e7·5-s + 1.44e9·7-s + 3.48e9·9-s + 1.01e11·11-s + 7.80e11·13-s + 1.49e12·15-s + 3.44e12·17-s − 2.35e13·19-s + 8.56e13·21-s + 2.44e13·23-s + 1.64e14·25-s + 2.05e14·27-s − 7.98e14·29-s − 2.09e15·31-s + 5.98e15·33-s + 3.67e16·35-s − 4.00e16·37-s + 4.61e16·39-s − 1.03e17·41-s − 2.02e17·43-s + 8.83e16·45-s − 6.52e17·47-s + 1.54e18·49-s + 2.03e17·51-s + 7.19e17·53-s + 2.56e18·55-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1.16·5-s + 1.94·7-s + 0.333·9-s + 1.17·11-s + 1.57·13-s + 0.669·15-s + 0.413·17-s − 0.881·19-s + 1.12·21-s + 0.123·23-s + 0.345·25-s + 0.192·27-s − 0.352·29-s − 0.459·31-s + 0.680·33-s + 2.25·35-s − 1.36·37-s + 0.906·39-s − 1.20·41-s − 1.42·43-s + 0.386·45-s − 1.80·47-s + 2.76·49-s + 0.239·51-s + 0.564·53-s + 1.36·55-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
| \(L(11)\) |
\(\approx\) |
\(4.941694582\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.941694582\) |
| \(L(\frac{23}{2})\) |
|
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 | \( 1 - 2.53e7T + 4.76e14T^{2} \) |
| 7 | \( 1 - 1.44e9T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.01e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 7.80e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 3.44e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 2.35e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 2.44e13T + 3.94e28T^{2} \) |
| 29 | \( 1 + 7.98e14T + 5.13e30T^{2} \) |
| 31 | \( 1 + 2.09e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.00e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 1.03e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.02e17T + 2.00e34T^{2} \) |
| 47 | \( 1 + 6.52e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 7.19e17T + 1.62e36T^{2} \) |
| 59 | \( 1 + 5.11e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 2.91e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 2.16e19T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.65e18T + 7.52e38T^{2} \) |
| 73 | \( 1 + 3.94e19T + 1.34e39T^{2} \) |
| 79 | \( 1 - 2.10e19T + 7.08e39T^{2} \) |
| 83 | \( 1 - 2.17e20T + 1.99e40T^{2} \) |
| 89 | \( 1 - 1.76e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 5.80e20T + 5.27e41T^{2} \) |
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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
−13.42987294442224596600540761655, −11.66954180465573952377203086287, −10.53616176521790789069003177479, −9.011128758526150905752310866511, −8.202847281696970620950001677149, −6.48090680874521262859276456712, −5.12253762421639653757111922168, −3.72094058930182633044603103482, −1.73379655959949628485345976515, −1.50277133174584964055773702453,
1.50277133174584964055773702453, 1.73379655959949628485345976515, 3.72094058930182633044603103482, 5.12253762421639653757111922168, 6.48090680874521262859276456712, 8.202847281696970620950001677149, 9.011128758526150905752310866511, 10.53616176521790789069003177479, 11.66954180465573952377203086287, 13.42987294442224596600540761655
