| L(s) = 1 | + 2.67·2-s − 1.26·3-s + 5.13·4-s − 0.362·5-s − 3.37·6-s + 0.0617·7-s + 8.38·8-s − 1.40·9-s − 0.968·10-s + 11-s − 6.48·12-s + 3.38·13-s + 0.165·14-s + 0.457·15-s + 12.1·16-s + 3.44·17-s − 3.75·18-s + 1.54·19-s − 1.86·20-s − 0.0779·21-s + 2.67·22-s + 6.24·23-s − 10.5·24-s − 4.86·25-s + 9.04·26-s + 5.56·27-s + 0.317·28-s + ⋯ |
| L(s) = 1 | + 1.88·2-s − 0.728·3-s + 2.56·4-s − 0.162·5-s − 1.37·6-s + 0.0233·7-s + 2.96·8-s − 0.468·9-s − 0.306·10-s + 0.301·11-s − 1.87·12-s + 0.939·13-s + 0.0441·14-s + 0.118·15-s + 3.03·16-s + 0.834·17-s − 0.885·18-s + 0.353·19-s − 0.416·20-s − 0.0170·21-s + 0.569·22-s + 1.30·23-s − 2.16·24-s − 0.973·25-s + 1.77·26-s + 1.07·27-s + 0.0599·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(4.478888520\) |
| \(L(\frac12)\) |
\(\approx\) |
\(4.478888520\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 11 | \( 1 - T \) |
| 131 | \( 1 + T \) |
| good | 2 | \( 1 - 2.67T + 2T^{2} \) |
| 3 | \( 1 + 1.26T + 3T^{2} \) |
| 5 | \( 1 + 0.362T + 5T^{2} \) |
| 7 | \( 1 - 0.0617T + 7T^{2} \) |
| 13 | \( 1 - 3.38T + 13T^{2} \) |
| 17 | \( 1 - 3.44T + 17T^{2} \) |
| 19 | \( 1 - 1.54T + 19T^{2} \) |
| 23 | \( 1 - 6.24T + 23T^{2} \) |
| 29 | \( 1 - 2.65T + 29T^{2} \) |
| 31 | \( 1 + 3.16T + 31T^{2} \) |
| 37 | \( 1 - 7.14T + 37T^{2} \) |
| 41 | \( 1 + 5.84T + 41T^{2} \) |
| 43 | \( 1 - 1.61T + 43T^{2} \) |
| 47 | \( 1 + 7.80T + 47T^{2} \) |
| 53 | \( 1 - 4.33T + 53T^{2} \) |
| 59 | \( 1 + 1.45T + 59T^{2} \) |
| 61 | \( 1 + 4.89T + 61T^{2} \) |
| 67 | \( 1 - 6.96T + 67T^{2} \) |
| 71 | \( 1 + 14.2T + 71T^{2} \) |
| 73 | \( 1 - 3.77T + 73T^{2} \) |
| 79 | \( 1 + 4.66T + 79T^{2} \) |
| 83 | \( 1 + 9.72T + 83T^{2} \) |
| 89 | \( 1 + 1.46T + 89T^{2} \) |
| 97 | \( 1 + 9.53T + 97T^{2} \) |
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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
−9.841564239066327514655277913275, −8.516230289110070608217028251713, −7.52544756961045933134333046973, −6.63405985991649492464055030741, −5.97953114277694517721890688538, −5.37745217893018735868324000137, −4.58870557920760831179363376842, −3.58560926999516488792442809686, −2.89381385060688321905020563684, −1.37482637578199326880136989974,
1.37482637578199326880136989974, 2.89381385060688321905020563684, 3.58560926999516488792442809686, 4.58870557920760831179363376842, 5.37745217893018735868324000137, 5.97953114277694517721890688538, 6.63405985991649492464055030741, 7.52544756961045933134333046973, 8.516230289110070608217028251713, 9.841564239066327514655277913275
