| L(s) = 1 | + 2.20·2-s + 2.88·4-s + 0.988·7-s + 1.94·8-s + 3.47·11-s − 5.78·13-s + 2.18·14-s − 1.46·16-s − 7.13·17-s + 6.26·19-s + 7.66·22-s + 1.36·23-s − 12.7·26-s + 2.84·28-s − 7.74·29-s − 9.06·31-s − 7.12·32-s − 15.7·34-s − 6.07·37-s + 13.8·38-s − 3.31·41-s + 43-s + 10.0·44-s + 3.02·46-s − 6.18·47-s − 6.02·49-s − 16.6·52-s + ⋯ |
| L(s) = 1 | + 1.56·2-s + 1.44·4-s + 0.373·7-s + 0.688·8-s + 1.04·11-s − 1.60·13-s + 0.583·14-s − 0.365·16-s − 1.72·17-s + 1.43·19-s + 1.63·22-s + 0.285·23-s − 2.50·26-s + 0.538·28-s − 1.43·29-s − 1.62·31-s − 1.25·32-s − 2.70·34-s − 0.999·37-s + 2.24·38-s − 0.517·41-s + 0.152·43-s + 1.50·44-s + 0.446·46-s − 0.901·47-s − 0.860·49-s − 2.31·52-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 9675 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | 3 | \( 1 \) |
| 5 | \( 1 \) |
| 43 | \( 1 - T \) |
| good | 2 | \( 1 - 2.20T + 2T^{2} \) |
| 7 | \( 1 - 0.988T + 7T^{2} \) |
| 11 | \( 1 - 3.47T + 11T^{2} \) |
| 13 | \( 1 + 5.78T + 13T^{2} \) |
| 17 | \( 1 + 7.13T + 17T^{2} \) |
| 19 | \( 1 - 6.26T + 19T^{2} \) |
| 23 | \( 1 - 1.36T + 23T^{2} \) |
| 29 | \( 1 + 7.74T + 29T^{2} \) |
| 31 | \( 1 + 9.06T + 31T^{2} \) |
| 37 | \( 1 + 6.07T + 37T^{2} \) |
| 41 | \( 1 + 3.31T + 41T^{2} \) |
| 47 | \( 1 + 6.18T + 47T^{2} \) |
| 53 | \( 1 + 0.867T + 53T^{2} \) |
| 59 | \( 1 + 0.261T + 59T^{2} \) |
| 61 | \( 1 - 13.5T + 61T^{2} \) |
| 67 | \( 1 - 0.229T + 67T^{2} \) |
| 71 | \( 1 + 12.2T + 71T^{2} \) |
| 73 | \( 1 - 2.02T + 73T^{2} \) |
| 79 | \( 1 - 9.66T + 79T^{2} \) |
| 83 | \( 1 - 1.04T + 83T^{2} \) |
| 89 | \( 1 - 6.15T + 89T^{2} \) |
| 97 | \( 1 + 1.10T + 97T^{2} \) |
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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
−7.08737925864557682041070555964, −6.64994378662701139014841161773, −5.73576627524186158158048657664, −5.06512329575502596457452134815, −4.73328328253585954447264761873, −3.83037507679105831350677337218, −3.33575225195025731598704058716, −2.29731006614570614063272107462, −1.72892470253258327486028388561, 0,
1.72892470253258327486028388561, 2.29731006614570614063272107462, 3.33575225195025731598704058716, 3.83037507679105831350677337218, 4.73328328253585954447264761873, 5.06512329575502596457452134815, 5.73576627524186158158048657664, 6.64994378662701139014841161773, 7.08737925864557682041070555964
