| L(s) = 1 | + 2.18·3-s − 3.70·7-s + 1.76·9-s − 3.18·11-s + 1.94·13-s − 4.66·17-s + 19-s − 8.08·21-s − 5.23·23-s − 2.70·27-s − 1.66·29-s − 3.46·31-s − 6.94·33-s + 7.18·37-s + 4.23·39-s − 0.717·41-s + 1.36·43-s − 5.37·47-s + 6.71·49-s − 10.1·51-s − 0.0435·53-s + 2.18·57-s − 7.88·59-s − 4.98·61-s − 6.52·63-s − 14.5·67-s − 11.4·69-s + ⋯ |
| L(s) = 1 | + 1.25·3-s − 1.39·7-s + 0.586·9-s − 0.959·11-s + 0.538·13-s − 1.13·17-s + 0.229·19-s − 1.76·21-s − 1.09·23-s − 0.520·27-s − 0.308·29-s − 0.622·31-s − 1.20·33-s + 1.18·37-s + 0.678·39-s − 0.112·41-s + 0.207·43-s − 0.784·47-s + 0.959·49-s − 1.42·51-s − 0.00597·53-s + 0.289·57-s − 1.02·59-s − 0.638·61-s − 0.821·63-s − 1.77·67-s − 1.37·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1900 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(=\) |
\(0\) |
| \(L(\frac12)\) |
\(=\) |
\(0\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 3 | \( 1 - 2.18T + 3T^{2} \) |
| 7 | \( 1 + 3.70T + 7T^{2} \) |
| 11 | \( 1 + 3.18T + 11T^{2} \) |
| 13 | \( 1 - 1.94T + 13T^{2} \) |
| 17 | \( 1 + 4.66T + 17T^{2} \) |
| 23 | \( 1 + 5.23T + 23T^{2} \) |
| 29 | \( 1 + 1.66T + 29T^{2} \) |
| 31 | \( 1 + 3.46T + 31T^{2} \) |
| 37 | \( 1 - 7.18T + 37T^{2} \) |
| 41 | \( 1 + 0.717T + 41T^{2} \) |
| 43 | \( 1 - 1.36T + 43T^{2} \) |
| 47 | \( 1 + 5.37T + 47T^{2} \) |
| 53 | \( 1 + 0.0435T + 53T^{2} \) |
| 59 | \( 1 + 7.88T + 59T^{2} \) |
| 61 | \( 1 + 4.98T + 61T^{2} \) |
| 67 | \( 1 + 14.5T + 67T^{2} \) |
| 71 | \( 1 - 4.88T + 71T^{2} \) |
| 73 | \( 1 + 10.6T + 73T^{2} \) |
| 79 | \( 1 - 14.9T + 79T^{2} \) |
| 83 | \( 1 + 9.76T + 83T^{2} \) |
| 89 | \( 1 - 10.0T + 89T^{2} \) |
| 97 | \( 1 - 14.6T + 97T^{2} \) |
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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
−8.968857537820316991092286185252, −8.048985526449180672080670061654, −7.48039322145522093271554322005, −6.43873843379510083771439849346, −5.80232781463336422858395235642, −4.46532343682410600224227115559, −3.53063917703608566567030418224, −2.88780524972653745712949156724, −2.01822135036020651751822802961, 0,
2.01822135036020651751822802961, 2.88780524972653745712949156724, 3.53063917703608566567030418224, 4.46532343682410600224227115559, 5.80232781463336422858395235642, 6.43873843379510083771439849346, 7.48039322145522093271554322005, 8.048985526449180672080670061654, 8.968857537820316991092286185252
