| L(s) = 1 | − 0.837·3-s − 2.01·5-s + 7-s − 2.29·9-s + 11-s + 13-s + 1.68·15-s − 4.49·17-s + 5.07·19-s − 0.837·21-s − 2.66·23-s − 0.928·25-s + 4.43·27-s − 1.39·29-s + 8.88·31-s − 0.837·33-s − 2.01·35-s − 5.57·37-s − 0.837·39-s − 1.42·41-s + 9.54·43-s + 4.63·45-s − 7.06·47-s + 49-s + 3.76·51-s − 2.80·53-s − 2.01·55-s + ⋯ |
| L(s) = 1 | − 0.483·3-s − 0.902·5-s + 0.377·7-s − 0.766·9-s + 0.301·11-s + 0.277·13-s + 0.436·15-s − 1.09·17-s + 1.16·19-s − 0.182·21-s − 0.556·23-s − 0.185·25-s + 0.854·27-s − 0.259·29-s + 1.59·31-s − 0.145·33-s − 0.341·35-s − 0.915·37-s − 0.134·39-s − 0.221·41-s + 1.45·43-s + 0.691·45-s − 1.03·47-s + 0.142·49-s + 0.527·51-s − 0.385·53-s − 0.272·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 8008 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 7 | \( 1 - T \) |
| 11 | \( 1 - T \) |
| 13 | \( 1 - T \) |
| good | 3 | \( 1 + 0.837T + 3T^{2} \) |
| 5 | \( 1 + 2.01T + 5T^{2} \) |
| 17 | \( 1 + 4.49T + 17T^{2} \) |
| 19 | \( 1 - 5.07T + 19T^{2} \) |
| 23 | \( 1 + 2.66T + 23T^{2} \) |
| 29 | \( 1 + 1.39T + 29T^{2} \) |
| 31 | \( 1 - 8.88T + 31T^{2} \) |
| 37 | \( 1 + 5.57T + 37T^{2} \) |
| 41 | \( 1 + 1.42T + 41T^{2} \) |
| 43 | \( 1 - 9.54T + 43T^{2} \) |
| 47 | \( 1 + 7.06T + 47T^{2} \) |
| 53 | \( 1 + 2.80T + 53T^{2} \) |
| 59 | \( 1 - 8.77T + 59T^{2} \) |
| 61 | \( 1 - 9.79T + 61T^{2} \) |
| 67 | \( 1 + 6.10T + 67T^{2} \) |
| 71 | \( 1 - 0.130T + 71T^{2} \) |
| 73 | \( 1 + 1.60T + 73T^{2} \) |
| 79 | \( 1 - 5.77T + 79T^{2} \) |
| 83 | \( 1 + 6.63T + 83T^{2} \) |
| 89 | \( 1 + 4.56T + 89T^{2} \) |
| 97 | \( 1 + 5.32T + 97T^{2} \) |
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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
−7.49446719498394158780512204095, −6.79717873889799280885583073281, −6.09939840488431655831922125909, −5.39387477275655985128474949615, −4.65422263453863415365512144876, −3.97636017961624092980913889962, −3.19301073882954530283910884442, −2.26569973128827656720548098553, −1.04508113307637760583549625555, 0,
1.04508113307637760583549625555, 2.26569973128827656720548098553, 3.19301073882954530283910884442, 3.97636017961624092980913889962, 4.65422263453863415365512144876, 5.39387477275655985128474949615, 6.09939840488431655831922125909, 6.79717873889799280885583073281, 7.49446719498394158780512204095
