| L(s) = 1 | + 3-s + 9-s − 3·11-s + 13-s − 7·19-s − 5·23-s + 27-s + 6·31-s − 3·33-s − 3·37-s + 39-s + 3·41-s + 8·43-s + 47-s − 5·53-s − 7·57-s − 4·59-s + 8·61-s − 5·69-s + 6·71-s − 14·73-s + 16·79-s + 81-s + 16·83-s − 6·89-s + 6·93-s + 16·97-s + ⋯ |
| L(s) = 1 | + 0.577·3-s + 1/3·9-s − 0.904·11-s + 0.277·13-s − 1.60·19-s − 1.04·23-s + 0.192·27-s + 1.07·31-s − 0.522·33-s − 0.493·37-s + 0.160·39-s + 0.468·41-s + 1.21·43-s + 0.145·47-s − 0.686·53-s − 0.927·57-s − 0.520·59-s + 1.02·61-s − 0.601·69-s + 0.712·71-s − 1.63·73-s + 1.80·79-s + 1/9·81-s + 1.75·83-s − 0.635·89-s + 0.622·93-s + 1.62·97-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 58800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
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 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 5 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 37 | \( 1 + 3 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 14 T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 16 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 - 16 T + p T^{2} \) |
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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
−14.57955247889153, −14.05398339041032, −13.58620690171007, −13.15388636292091, −12.52225457989181, −12.30135408401430, −11.50697065008264, −10.90122883060256, −10.43602563626131, −10.10904977501148, −9.360209011441947, −8.909059807505478, −8.234782906619250, −8.001918321340518, −7.426445939327318, −6.644860964588996, −6.216955475616820, −5.629111242055402, −4.846784487106846, −4.330941396444921, −3.782156724526464, −3.058476887834843, −2.326218990148043, −2.008419252823793, −0.9350577989650694, 0,
0.9350577989650694, 2.008419252823793, 2.326218990148043, 3.058476887834843, 3.782156724526464, 4.330941396444921, 4.846784487106846, 5.629111242055402, 6.216955475616820, 6.644860964588996, 7.426445939327318, 8.001918321340518, 8.234782906619250, 8.909059807505478, 9.360209011441947, 10.10904977501148, 10.43602563626131, 10.90122883060256, 11.50697065008264, 12.30135408401430, 12.52225457989181, 13.15388636292091, 13.58620690171007, 14.05398339041032, 14.57955247889153
