| L(s) = 1 | + 2.42·2-s − 2.86·3-s + 3.89·4-s + 0.116·5-s − 6.94·6-s + 2.69·7-s + 4.58·8-s + 5.19·9-s + 0.281·10-s − 3.38·11-s − 11.1·12-s + 1.05·13-s + 6.55·14-s − 0.332·15-s + 3.35·16-s + 3.65·17-s + 12.5·18-s + 2.27·19-s + 0.451·20-s − 7.72·21-s − 8.21·22-s + 7.49·23-s − 13.1·24-s − 4.98·25-s + 2.55·26-s − 6.26·27-s + 10.5·28-s + ⋯ |
| L(s) = 1 | + 1.71·2-s − 1.65·3-s + 1.94·4-s + 0.0519·5-s − 2.83·6-s + 1.02·7-s + 1.62·8-s + 1.73·9-s + 0.0891·10-s − 1.02·11-s − 3.21·12-s + 0.291·13-s + 1.75·14-s − 0.0857·15-s + 0.838·16-s + 0.886·17-s + 2.96·18-s + 0.523·19-s + 0.100·20-s − 1.68·21-s − 1.75·22-s + 1.56·23-s − 2.68·24-s − 0.997·25-s + 0.500·26-s − 1.20·27-s + 1.98·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1681 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.188628160\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.188628160\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 41 | \( 1 \) |
| good | 2 | \( 1 - 2.42T + 2T^{2} \) |
| 3 | \( 1 + 2.86T + 3T^{2} \) |
| 5 | \( 1 - 0.116T + 5T^{2} \) |
| 7 | \( 1 - 2.69T + 7T^{2} \) |
| 11 | \( 1 + 3.38T + 11T^{2} \) |
| 13 | \( 1 - 1.05T + 13T^{2} \) |
| 17 | \( 1 - 3.65T + 17T^{2} \) |
| 19 | \( 1 - 2.27T + 19T^{2} \) |
| 23 | \( 1 - 7.49T + 23T^{2} \) |
| 29 | \( 1 - 1.47T + 29T^{2} \) |
| 31 | \( 1 - 3.12T + 31T^{2} \) |
| 37 | \( 1 + 1.12T + 37T^{2} \) |
| 43 | \( 1 - 0.993T + 43T^{2} \) |
| 47 | \( 1 - 11.0T + 47T^{2} \) |
| 53 | \( 1 + 0.542T + 53T^{2} \) |
| 59 | \( 1 - 1.94T + 59T^{2} \) |
| 61 | \( 1 - 10.1T + 61T^{2} \) |
| 67 | \( 1 - 13.2T + 67T^{2} \) |
| 71 | \( 1 - 9.43T + 71T^{2} \) |
| 73 | \( 1 - 10.9T + 73T^{2} \) |
| 79 | \( 1 + 10.7T + 79T^{2} \) |
| 83 | \( 1 + 0.635T + 83T^{2} \) |
| 89 | \( 1 + 4.81T + 89T^{2} \) |
| 97 | \( 1 + 2.76T + 97T^{2} \) |
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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
−9.753505482703774406162230731093, −8.196299730679505071244074490297, −7.31408417777121815192212369761, −6.61092775949325344697892763539, −5.55206138911157226338534139100, −5.37977424665998353351115893672, −4.72860578849097234593203797974, −3.78248220124778880787109147891, −2.52266020931341021088298664989, −1.12005235412749766677302132890,
1.12005235412749766677302132890, 2.52266020931341021088298664989, 3.78248220124778880787109147891, 4.72860578849097234593203797974, 5.37977424665998353351115893672, 5.55206138911157226338534139100, 6.61092775949325344697892763539, 7.31408417777121815192212369761, 8.196299730679505071244074490297, 9.753505482703774406162230731093
