| L(s) = 1 | − 5-s + 2·7-s − 3·9-s + 4·13-s + 4·17-s + 25-s + 6·29-s − 2·35-s − 2·37-s − 6·41-s − 2·43-s + 3·45-s − 3·49-s − 10·53-s + 12·59-s + 6·61-s − 6·63-s − 4·65-s − 12·67-s + 16·71-s − 4·73-s + 4·79-s + 9·81-s − 2·83-s − 4·85-s + 6·89-s + 8·91-s + ⋯ |
| L(s) = 1 | − 0.447·5-s + 0.755·7-s − 9-s + 1.10·13-s + 0.970·17-s + 1/5·25-s + 1.11·29-s − 0.338·35-s − 0.328·37-s − 0.937·41-s − 0.304·43-s + 0.447·45-s − 3/7·49-s − 1.37·53-s + 1.56·59-s + 0.768·61-s − 0.755·63-s − 0.496·65-s − 1.46·67-s + 1.89·71-s − 0.468·73-s + 0.450·79-s + 81-s − 0.219·83-s − 0.433·85-s + 0.635·89-s + 0.838·91-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4840 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.892329547\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.892329547\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 + T \) |
| 11 | \( 1 \) |
| good | 3 | \( 1 + p T^{2} \) |
| 7 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 19 | \( 1 + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 - 12 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 12 T + p T^{2} \) |
| 71 | \( 1 - 16 T + p T^{2} \) |
| 73 | \( 1 + 4 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 2 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 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
−8.378504306303230595792573421898, −7.76028902701821629189652051955, −6.80570962877372632758672621835, −6.06624510237246593342421161666, −5.32238319845644139561140980648, −4.65007191464064765726623390167, −3.61023365236105810017752942139, −3.05615336219741886208723963694, −1.83947258552734013700314787947, −0.77727709017353821586723677306,
0.77727709017353821586723677306, 1.83947258552734013700314787947, 3.05615336219741886208723963694, 3.61023365236105810017752942139, 4.65007191464064765726623390167, 5.32238319845644139561140980648, 6.06624510237246593342421161666, 6.80570962877372632758672621835, 7.76028902701821629189652051955, 8.378504306303230595792573421898
