| L(s) = 1 | − 2·7-s + 13-s + 17-s + 8·19-s + 6·23-s + 7·29-s + 3·31-s + 2·37-s + 12·41-s + 8·43-s − 6·47-s − 3·49-s − 9·53-s − 9·59-s + 2·61-s − 67-s − 2·71-s − 2·73-s − 6·79-s + 12·83-s − 89-s − 2·91-s − 4·97-s + 101-s + 103-s + 107-s + 109-s + ⋯ |
| L(s) = 1 | − 0.755·7-s + 0.277·13-s + 0.242·17-s + 1.83·19-s + 1.25·23-s + 1.29·29-s + 0.538·31-s + 0.328·37-s + 1.87·41-s + 1.21·43-s − 0.875·47-s − 3/7·49-s − 1.23·53-s − 1.17·59-s + 0.256·61-s − 0.122·67-s − 0.237·71-s − 0.234·73-s − 0.675·79-s + 1.31·83-s − 0.105·89-s − 0.209·91-s − 0.406·97-s + 0.0995·101-s + 0.0985·103-s + 0.0966·107-s + 0.0957·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 397800 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.062011126\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.062011126\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| 13 | \( 1 - T \) |
| 17 | \( 1 - T \) |
| good | 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 8 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 - 7 T + p T^{2} \) |
| 31 | \( 1 - 3 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 12 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 9 T + p T^{2} \) |
| 59 | \( 1 + 9 T + p T^{2} \) |
| 61 | \( 1 - 2 T + p T^{2} \) |
| 67 | \( 1 + T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + T + p T^{2} \) |
| 97 | \( 1 + 4 T + p T^{2} \) |
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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
−12.46586855616870, −12.00298423861138, −11.58026602835247, −11.02567498102914, −10.72444372505263, −10.14492804161084, −9.591763460558755, −9.342958951751219, −9.046741922067933, −8.214734264492319, −7.919150394535738, −7.387813819339338, −6.964092116254019, −6.363525026574557, −6.078944855941612, −5.480084382174023, −4.949864002836018, −4.535621762025264, −3.887912659308609, −3.277557156411874, −2.845710843560881, −2.623171951634170, −1.474247924398358, −1.092744880544564, −0.5076240489910886,
0.5076240489910886, 1.092744880544564, 1.474247924398358, 2.623171951634170, 2.845710843560881, 3.277557156411874, 3.887912659308609, 4.535621762025264, 4.949864002836018, 5.480084382174023, 6.078944855941612, 6.363525026574557, 6.964092116254019, 7.387813819339338, 7.919150394535738, 8.214734264492319, 9.046741922067933, 9.342958951751219, 9.591763460558755, 10.14492804161084, 10.72444372505263, 11.02567498102914, 11.58026602835247, 12.00298423861138, 12.46586855616870
