| L(s) = 1 | + 2·5-s − 7-s − 6·11-s + 6·13-s + 2·17-s + 4·19-s + 2·23-s − 25-s + 8·29-s − 4·31-s − 2·35-s + 6·37-s − 10·41-s − 4·43-s − 4·47-s + 49-s − 4·53-s − 12·55-s + 12·59-s + 2·61-s + 12·65-s + 12·67-s + 6·71-s − 2·73-s + 6·77-s + 8·79-s + 4·85-s + ⋯ |
| L(s) = 1 | + 0.894·5-s − 0.377·7-s − 1.80·11-s + 1.66·13-s + 0.485·17-s + 0.917·19-s + 0.417·23-s − 1/5·25-s + 1.48·29-s − 0.718·31-s − 0.338·35-s + 0.986·37-s − 1.56·41-s − 0.609·43-s − 0.583·47-s + 1/7·49-s − 0.549·53-s − 1.61·55-s + 1.56·59-s + 0.256·61-s + 1.48·65-s + 1.46·67-s + 0.712·71-s − 0.234·73-s + 0.683·77-s + 0.900·79-s + 0.433·85-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4032 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.174938603\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.174938603\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 2 T + p T^{2} \) |
| 19 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 - 8 T + p T^{2} \) |
| 31 | \( 1 + 4 T + p T^{2} \) |
| 37 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + 10 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 + 4 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 - 12 T + p T^{2} \) |
| 71 | \( 1 - 6 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 - 14 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.330422685772202607346193368012, −7.898186089886958744036095771574, −6.84845331752847395734818805677, −6.17874926148896028530358602827, −5.44238850248002906452836994978, −4.98128143723083841306992002949, −3.61651253641666745721285322081, −2.98026092932185622382993880012, −2.00703502347559203436322652938, −0.857404457451122055017254113328,
0.857404457451122055017254113328, 2.00703502347559203436322652938, 2.98026092932185622382993880012, 3.61651253641666745721285322081, 4.98128143723083841306992002949, 5.44238850248002906452836994978, 6.17874926148896028530358602827, 6.84845331752847395734818805677, 7.898186089886958744036095771574, 8.330422685772202607346193368012
