L(s) = 1 | + 2·3-s + 5-s + 9-s + 4·11-s + 2·13-s + 2·15-s + 2·19-s − 4·23-s + 25-s − 4·27-s + 10·29-s + 4·31-s + 8·33-s − 2·37-s + 4·39-s − 12·41-s − 4·43-s + 45-s + 4·47-s + 2·53-s + 4·55-s + 4·57-s + 10·59-s + 6·61-s + 2·65-s + 4·67-s − 8·69-s + ⋯ |
L(s) = 1 | + 1.15·3-s + 0.447·5-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 0.516·15-s + 0.458·19-s − 0.834·23-s + 1/5·25-s − 0.769·27-s + 1.85·29-s + 0.718·31-s + 1.39·33-s − 0.328·37-s + 0.640·39-s − 1.87·41-s − 0.609·43-s + 0.149·45-s + 0.583·47-s + 0.274·53-s + 0.539·55-s + 0.529·57-s + 1.30·59-s + 0.768·61-s + 0.248·65-s + 0.488·67-s − 0.963·69-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1960 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.112396210\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.112396210\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 - T \) |
| 7 | \( 1 \) |
good | 3 | \( 1 - 2 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 4 T + p T^{2} \) |
| 29 | \( 1 - 10 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 12 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 + 12 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 + 4 T + p T^{2} \) |
| 83 | \( 1 - 14 T + p T^{2} \) |
| 89 | \( 1 - 8 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−8.879865708970217822824383652965, −8.658241163787682608808024071308, −7.82224725183302503327537705396, −6.76696800496424630692002767552, −6.19923277418465196349162677788, −5.08442527511306327851298395873, −3.98912322632220726154350409435, −3.28931589292673252560971877014, −2.30210104956127010785041178683, −1.25496120705537884599858448710,
1.25496120705537884599858448710, 2.30210104956127010785041178683, 3.28931589292673252560971877014, 3.98912322632220726154350409435, 5.08442527511306327851298395873, 6.19923277418465196349162677788, 6.76696800496424630692002767552, 7.82224725183302503327537705396, 8.658241163787682608808024071308, 8.879865708970217822824383652965