| L(s) = 1 | − 4.79e4·3-s − 9.76e6·5-s − 6.67e8·7-s − 8.16e9·9-s + 1.26e11·11-s + 9.12e11·13-s + 4.68e11·15-s + 8.60e12·17-s + 6.93e12·19-s + 3.20e13·21-s + 3.30e14·23-s + 9.53e13·25-s + 8.92e14·27-s − 3.90e15·29-s − 3.32e15·31-s − 6.07e15·33-s + 6.52e15·35-s − 4.33e16·37-s − 4.37e16·39-s − 9.56e16·41-s + 7.81e16·43-s + 7.97e16·45-s + 2.03e17·47-s − 1.12e17·49-s − 4.12e17·51-s + 1.42e18·53-s − 1.23e18·55-s + ⋯ |
| L(s) = 1 | − 0.468·3-s − 0.447·5-s − 0.893·7-s − 0.780·9-s + 1.47·11-s + 1.83·13-s + 0.209·15-s + 1.03·17-s + 0.259·19-s + 0.418·21-s + 1.66·23-s + 0.199·25-s + 0.834·27-s − 1.72·29-s − 0.727·31-s − 0.690·33-s + 0.399·35-s − 1.48·37-s − 0.860·39-s − 1.11·41-s + 0.551·43-s + 0.349·45-s + 0.564·47-s − 0.201·49-s − 0.485·51-s + 1.11·53-s − 0.659·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(11)\) |
\(\approx\) |
\(1.688782996\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.688782996\) |
| \(L(\frac{23}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + 9.76e6T \) |
| good | 3 | \( 1 + 4.79e4T + 1.04e10T^{2} \) |
| 7 | \( 1 + 6.67e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 1.26e11T + 7.40e21T^{2} \) |
| 13 | \( 1 - 9.12e11T + 2.47e23T^{2} \) |
| 17 | \( 1 - 8.60e12T + 6.90e25T^{2} \) |
| 19 | \( 1 - 6.93e12T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.30e14T + 3.94e28T^{2} \) |
| 29 | \( 1 + 3.90e15T + 5.13e30T^{2} \) |
| 31 | \( 1 + 3.32e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 4.33e16T + 8.55e32T^{2} \) |
| 41 | \( 1 + 9.56e16T + 7.38e33T^{2} \) |
| 43 | \( 1 - 7.81e16T + 2.00e34T^{2} \) |
| 47 | \( 1 - 2.03e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.42e18T + 1.62e36T^{2} \) |
| 59 | \( 1 + 2.28e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.32e18T + 3.10e37T^{2} \) |
| 67 | \( 1 - 1.31e19T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.98e19T + 7.52e38T^{2} \) |
| 73 | \( 1 - 1.37e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 1.74e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 3.75e19T + 1.99e40T^{2} \) |
| 89 | \( 1 - 2.43e20T + 8.65e40T^{2} \) |
| 97 | \( 1 + 1.80e20T + 5.27e41T^{2} \) |
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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
−10.78864462770805101729053899041, −9.286662644454206807728707160482, −8.644865390586306989862122776662, −7.12621932811982775529013102149, −6.23205311757413509889144840701, −5.37721946921374744253736107503, −3.66545286488089139818453427330, −3.35006483537657709078053752191, −1.44926322757208746880444991362, −0.59413376800117591162270982246,
0.59413376800117591162270982246, 1.44926322757208746880444991362, 3.35006483537657709078053752191, 3.66545286488089139818453427330, 5.37721946921374744253736107503, 6.23205311757413509889144840701, 7.12621932811982775529013102149, 8.644865390586306989862122776662, 9.286662644454206807728707160482, 10.78864462770805101729053899041
