| L(s) = 1 | − 1.93·2-s − 3-s + 1.76·4-s + 3.07·5-s + 1.93·6-s − 3.15·7-s + 0.464·8-s + 9-s − 5.96·10-s − 11-s − 1.76·12-s − 4.31·13-s + 6.12·14-s − 3.07·15-s − 4.42·16-s − 3.81·17-s − 1.93·18-s − 4.23·19-s + 5.41·20-s + 3.15·21-s + 1.93·22-s + 6.48·23-s − 0.464·24-s + 4.47·25-s + 8.35·26-s − 27-s − 5.56·28-s + ⋯ |
| L(s) = 1 | − 1.37·2-s − 0.577·3-s + 0.880·4-s + 1.37·5-s + 0.791·6-s − 1.19·7-s + 0.164·8-s + 0.333·9-s − 1.88·10-s − 0.301·11-s − 0.508·12-s − 1.19·13-s + 1.63·14-s − 0.794·15-s − 1.10·16-s − 0.924·17-s − 0.457·18-s − 0.971·19-s + 1.21·20-s + 0.689·21-s + 0.413·22-s + 1.35·23-s − 0.0947·24-s + 0.894·25-s + 1.63·26-s − 0.192·27-s − 1.05·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2013 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.5334221993\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.5334221993\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + T \) |
| 11 | \( 1 + T \) |
| 61 | \( 1 - T \) |
| good | 2 | \( 1 + 1.93T + 2T^{2} \) |
| 5 | \( 1 - 3.07T + 5T^{2} \) |
| 7 | \( 1 + 3.15T + 7T^{2} \) |
| 13 | \( 1 + 4.31T + 13T^{2} \) |
| 17 | \( 1 + 3.81T + 17T^{2} \) |
| 19 | \( 1 + 4.23T + 19T^{2} \) |
| 23 | \( 1 - 6.48T + 23T^{2} \) |
| 29 | \( 1 + 3.27T + 29T^{2} \) |
| 31 | \( 1 - 10.0T + 31T^{2} \) |
| 37 | \( 1 + 1.85T + 37T^{2} \) |
| 41 | \( 1 + 10.7T + 41T^{2} \) |
| 43 | \( 1 - 10.4T + 43T^{2} \) |
| 47 | \( 1 - 10.3T + 47T^{2} \) |
| 53 | \( 1 - 8.28T + 53T^{2} \) |
| 59 | \( 1 + 10.0T + 59T^{2} \) |
| 67 | \( 1 + 5.80T + 67T^{2} \) |
| 71 | \( 1 + 3.40T + 71T^{2} \) |
| 73 | \( 1 - 11.2T + 73T^{2} \) |
| 79 | \( 1 - 4.26T + 79T^{2} \) |
| 83 | \( 1 - 17.4T + 83T^{2} \) |
| 89 | \( 1 + 9.75T + 89T^{2} \) |
| 97 | \( 1 - 0.814T + 97T^{2} \) |
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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
−9.249257876790318779662072149132, −8.766638329744803801825007616207, −7.54912134687772359592525331152, −6.75079329407211919260897385265, −6.35006556302158460755366733601, −5.30896408543717654534408909000, −4.44413363798550950608434846348, −2.73630513729625009361381992786, −2.01506999587665040740849435398, −0.59302954548388644906828221418,
0.59302954548388644906828221418, 2.01506999587665040740849435398, 2.73630513729625009361381992786, 4.44413363798550950608434846348, 5.30896408543717654534408909000, 6.35006556302158460755366733601, 6.75079329407211919260897385265, 7.54912134687772359592525331152, 8.766638329744803801825007616207, 9.249257876790318779662072149132
