| L(s) = 1 | − 1.67e5·3-s + 3.35e6·5-s + 7.07e8·7-s + 1.76e10·9-s + 8.75e10·11-s − 7.41e11·13-s − 5.62e11·15-s − 6.82e12·17-s + 5.17e13·19-s − 1.18e14·21-s − 3.13e14·23-s − 4.65e14·25-s − 1.20e15·27-s + 1.46e15·29-s − 6.42e15·31-s − 1.46e16·33-s + 2.37e15·35-s − 6.93e15·37-s + 1.24e17·39-s + 3.25e16·41-s + 4.37e16·43-s + 5.92e16·45-s − 5.34e16·47-s − 5.75e16·49-s + 1.14e18·51-s − 1.19e18·53-s + 2.93e17·55-s + ⋯ |
| L(s) = 1 | − 1.63·3-s + 0.153·5-s + 0.947·7-s + 1.68·9-s + 1.01·11-s − 1.49·13-s − 0.251·15-s − 0.821·17-s + 1.93·19-s − 1.55·21-s − 1.57·23-s − 0.976·25-s − 1.12·27-s + 0.646·29-s − 1.40·31-s − 1.66·33-s + 0.145·35-s − 0.237·37-s + 2.44·39-s + 0.378·41-s + 0.308·43-s + 0.259·45-s − 0.148·47-s − 0.103·49-s + 1.34·51-s − 0.936·53-s + 0.156·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| good | 3 | \( 1 + 1.67e5T + 1.04e10T^{2} \) |
| 5 | \( 1 - 3.35e6T + 4.76e14T^{2} \) |
| 7 | \( 1 - 7.07e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 8.75e10T + 7.40e21T^{2} \) |
| 13 | \( 1 + 7.41e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 6.82e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 5.17e13T + 7.14e26T^{2} \) |
| 23 | \( 1 + 3.13e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 1.46e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 6.42e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 6.93e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 3.25e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 4.37e16T + 2.00e34T^{2} \) |
| 47 | \( 1 + 5.34e16T + 1.30e35T^{2} \) |
| 53 | \( 1 + 1.19e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 4.48e17T + 1.54e37T^{2} \) |
| 61 | \( 1 + 9.05e17T + 3.10e37T^{2} \) |
| 67 | \( 1 + 6.06e17T + 2.22e38T^{2} \) |
| 71 | \( 1 + 2.18e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 6.65e19T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.81e18T + 7.08e39T^{2} \) |
| 83 | \( 1 + 2.58e20T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.80e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 4.09e20T + 5.27e41T^{2} \) |
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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
−16.07738923126480105880119040042, −14.25211331703137055786635219295, −12.12523938267164376204454857923, −11.40606348352907274673977334056, −9.788088767360654514273580015730, −7.32114213026982426967722573292, −5.73288270878084073572771322557, −4.52580863131340504127186642360, −1.58045383595620525973313110003, 0,
1.58045383595620525973313110003, 4.52580863131340504127186642360, 5.73288270878084073572771322557, 7.32114213026982426967722573292, 9.788088767360654514273580015730, 11.40606348352907274673977334056, 12.12523938267164376204454857923, 14.25211331703137055786635219295, 16.07738923126480105880119040042
