| L(s) = 1 | + 3·3-s + 5-s + 2·7-s + 6·9-s + 13-s + 3·15-s + 6·21-s − 23-s + 25-s + 9·27-s − 3·29-s − 3·31-s + 2·35-s − 8·37-s + 3·39-s + 3·41-s + 2·43-s + 6·45-s + 11·47-s − 3·49-s − 14·53-s + 8·59-s − 4·61-s + 12·63-s + 65-s + 4·67-s − 3·69-s + ⋯ |
| L(s) = 1 | + 1.73·3-s + 0.447·5-s + 0.755·7-s + 2·9-s + 0.277·13-s + 0.774·15-s + 1.30·21-s − 0.208·23-s + 1/5·25-s + 1.73·27-s − 0.557·29-s − 0.538·31-s + 0.338·35-s − 1.31·37-s + 0.480·39-s + 0.468·41-s + 0.304·43-s + 0.894·45-s + 1.60·47-s − 3/7·49-s − 1.92·53-s + 1.04·59-s − 0.512·61-s + 1.51·63-s + 0.124·65-s + 0.488·67-s − 0.361·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.871277106\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.871277106\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 23 | \( 1 + T \) |
| good | 3 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 + 3 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 2 T + p T^{2} \) |
| 47 | \( 1 - 11 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 7 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 + 4 T + p T^{2} \) |
| 89 | \( 1 + 2 T + p T^{2} \) |
| 97 | \( 1 - 18 T + p T^{2} \) |
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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
−8.999051054088738205599213223507, −8.642854744976674671107691765539, −7.73854228255682207536319172149, −7.26632068670665245818970027733, −6.10696554539960738301895722103, −5.02356588927784018754647193422, −4.07459093437266598889393557689, −3.25036929536985039826813471791, −2.24637242067700269920582104471, −1.50488623951800161308613033156,
1.50488623951800161308613033156, 2.24637242067700269920582104471, 3.25036929536985039826813471791, 4.07459093437266598889393557689, 5.02356588927784018754647193422, 6.10696554539960738301895722103, 7.26632068670665245818970027733, 7.73854228255682207536319172149, 8.642854744976674671107691765539, 8.999051054088738205599213223507
