| L(s) = 1 | + 0.269·2-s + 1.39·3-s − 1.92·4-s + 3.37·5-s + 0.377·6-s − 1.05·8-s − 1.04·9-s + 0.911·10-s − 1.57·11-s − 2.69·12-s + 2.85·13-s + 4.72·15-s + 3.56·16-s + 6.37·17-s − 0.281·18-s + 19-s − 6.51·20-s − 0.426·22-s + 2.28·23-s − 1.48·24-s + 6.42·25-s + 0.768·26-s − 5.65·27-s + 8.17·29-s + 1.27·30-s + 4.99·31-s + 3.08·32-s + ⋯ |
| L(s) = 1 | + 0.190·2-s + 0.807·3-s − 0.963·4-s + 1.51·5-s + 0.153·6-s − 0.374·8-s − 0.347·9-s + 0.288·10-s − 0.476·11-s − 0.778·12-s + 0.790·13-s + 1.22·15-s + 0.892·16-s + 1.54·17-s − 0.0663·18-s + 0.229·19-s − 1.45·20-s − 0.0908·22-s + 0.476·23-s − 0.302·24-s + 1.28·25-s + 0.150·26-s − 1.08·27-s + 1.51·29-s + 0.232·30-s + 0.897·31-s + 0.544·32-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 931 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.366919637\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.366919637\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 2 | \( 1 - 0.269T + 2T^{2} \) |
| 3 | \( 1 - 1.39T + 3T^{2} \) |
| 5 | \( 1 - 3.37T + 5T^{2} \) |
| 11 | \( 1 + 1.57T + 11T^{2} \) |
| 13 | \( 1 - 2.85T + 13T^{2} \) |
| 17 | \( 1 - 6.37T + 17T^{2} \) |
| 23 | \( 1 - 2.28T + 23T^{2} \) |
| 29 | \( 1 - 8.17T + 29T^{2} \) |
| 31 | \( 1 - 4.99T + 31T^{2} \) |
| 37 | \( 1 + 10.6T + 37T^{2} \) |
| 41 | \( 1 + 0.539T + 41T^{2} \) |
| 43 | \( 1 - 2.79T + 43T^{2} \) |
| 47 | \( 1 + 9.04T + 47T^{2} \) |
| 53 | \( 1 - 2.71T + 53T^{2} \) |
| 59 | \( 1 + 9.03T + 59T^{2} \) |
| 61 | \( 1 - 6.41T + 61T^{2} \) |
| 67 | \( 1 - 0.976T + 67T^{2} \) |
| 71 | \( 1 + 10.9T + 71T^{2} \) |
| 73 | \( 1 - 3.06T + 73T^{2} \) |
| 79 | \( 1 - 0.692T + 79T^{2} \) |
| 83 | \( 1 - 4.23T + 83T^{2} \) |
| 89 | \( 1 + 4.04T + 89T^{2} \) |
| 97 | \( 1 - 15.0T + 97T^{2} \) |
| 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
−10.00731997787558064578196478377, −9.167485274989695769000957317191, −8.571329368565386512491569715199, −7.85599307763441909386729719071, −6.40737789411643466535185995336, −5.59250484536517097139475141793, −4.96162474717281920291942067049, −3.49326078377733167600625703915, −2.76434065386641114082434410177, −1.30815701203702388791937828467,
1.30815701203702388791937828467, 2.76434065386641114082434410177, 3.49326078377733167600625703915, 4.96162474717281920291942067049, 5.59250484536517097139475141793, 6.40737789411643466535185995336, 7.85599307763441909386729719071, 8.571329368565386512491569715199, 9.167485274989695769000957317191, 10.00731997787558064578196478377
