| L(s) = 1 | − 1.83e4·3-s − 3.58e7·5-s − 9.75e8·7-s − 1.01e10·9-s + 5.98e10·11-s + 4.80e11·13-s + 6.59e11·15-s − 7.70e12·17-s − 5.11e13·19-s + 1.79e13·21-s + 3.46e14·23-s + 8.08e14·25-s + 3.78e14·27-s + 2.04e15·29-s + 2.31e15·31-s − 1.10e15·33-s + 3.49e16·35-s − 2.31e15·37-s − 8.83e15·39-s + 1.49e17·41-s − 2.47e17·43-s + 3.62e17·45-s + 1.64e17·47-s + 3.93e17·49-s + 1.41e17·51-s + 1.10e18·53-s − 2.14e18·55-s + ⋯ |
| L(s) = 1 | − 0.179·3-s − 1.64·5-s − 1.30·7-s − 0.967·9-s + 0.695·11-s + 0.966·13-s + 0.295·15-s − 0.926·17-s − 1.91·19-s + 0.234·21-s + 1.74·23-s + 1.69·25-s + 0.353·27-s + 0.902·29-s + 0.508·31-s − 0.125·33-s + 2.14·35-s − 0.0792·37-s − 0.173·39-s + 1.73·41-s − 1.74·43-s + 1.58·45-s + 0.455·47-s + 0.703·49-s + 0.166·51-s + 0.867·53-s − 1.14·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(22-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 64 ^{s/2} \, \Gamma_{\C}(s+21/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 + 1.83e4T + 1.04e10T^{2} \) |
| 5 | \( 1 + 3.58e7T + 4.76e14T^{2} \) |
| 7 | \( 1 + 9.75e8T + 5.58e17T^{2} \) |
| 11 | \( 1 - 5.98e10T + 7.40e21T^{2} \) |
| 13 | \( 1 - 4.80e11T + 2.47e23T^{2} \) |
| 17 | \( 1 + 7.70e12T + 6.90e25T^{2} \) |
| 19 | \( 1 + 5.11e13T + 7.14e26T^{2} \) |
| 23 | \( 1 - 3.46e14T + 3.94e28T^{2} \) |
| 29 | \( 1 - 2.04e15T + 5.13e30T^{2} \) |
| 31 | \( 1 - 2.31e15T + 2.08e31T^{2} \) |
| 37 | \( 1 + 2.31e15T + 8.55e32T^{2} \) |
| 41 | \( 1 - 1.49e17T + 7.38e33T^{2} \) |
| 43 | \( 1 + 2.47e17T + 2.00e34T^{2} \) |
| 47 | \( 1 - 1.64e17T + 1.30e35T^{2} \) |
| 53 | \( 1 - 1.10e18T + 1.62e36T^{2} \) |
| 59 | \( 1 - 2.32e18T + 1.54e37T^{2} \) |
| 61 | \( 1 + 5.91e18T + 3.10e37T^{2} \) |
| 67 | \( 1 + 4.24e18T + 2.22e38T^{2} \) |
| 71 | \( 1 - 2.52e19T + 7.52e38T^{2} \) |
| 73 | \( 1 + 5.77e18T + 1.34e39T^{2} \) |
| 79 | \( 1 + 3.50e19T + 7.08e39T^{2} \) |
| 83 | \( 1 + 3.66e19T + 1.99e40T^{2} \) |
| 89 | \( 1 + 1.12e20T + 8.65e40T^{2} \) |
| 97 | \( 1 - 6.65e20T + 5.27e41T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−10.71863515154582262447359338953, −8.947257856699596756910447931755, −8.428124128394472379727550276366, −6.86590377347468343504509966417, −6.25662026611195865689513660656, −4.50547123124074334295061461111, −3.63263750258691305197543700990, −2.74621510774570886100916111333, −0.78673750526868555665160197868, 0,
0.78673750526868555665160197868, 2.74621510774570886100916111333, 3.63263750258691305197543700990, 4.50547123124074334295061461111, 6.25662026611195865689513660656, 6.86590377347468343504509966417, 8.428124128394472379727550276366, 8.947257856699596756910447931755, 10.71863515154582262447359338953
