| L(s) = 1 | + 5-s + 3·7-s − 5·11-s + 4·13-s + 8·17-s + 2·19-s − 2·23-s − 4·25-s − 6·29-s − 7·31-s + 3·35-s − 6·37-s + 6·41-s − 2·43-s − 6·47-s + 2·49-s − 5·53-s − 5·55-s + 4·59-s − 8·61-s + 4·65-s − 10·67-s + 8·71-s + 73-s − 15·77-s + 16·79-s + 11·83-s + ⋯ |
| L(s) = 1 | + 0.447·5-s + 1.13·7-s − 1.50·11-s + 1.10·13-s + 1.94·17-s + 0.458·19-s − 0.417·23-s − 4/5·25-s − 1.11·29-s − 1.25·31-s + 0.507·35-s − 0.986·37-s + 0.937·41-s − 0.304·43-s − 0.875·47-s + 2/7·49-s − 0.686·53-s − 0.674·55-s + 0.520·59-s − 1.02·61-s + 0.496·65-s − 1.22·67-s + 0.949·71-s + 0.117·73-s − 1.70·77-s + 1.80·79-s + 1.20·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.389247429\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.389247429\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 - T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 23 | \( 1 + 2 T + p T^{2} \) |
| 29 | \( 1 + 6 T + p T^{2} \) |
| 31 | \( 1 + 7 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 - 6 T + p T^{2} \) |
| 43 | \( 1 + 2 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 5 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 10 T + p T^{2} \) |
| 71 | \( 1 - 8 T + p T^{2} \) |
| 73 | \( 1 - T + p T^{2} \) |
| 79 | \( 1 - 16 T + p T^{2} \) |
| 83 | \( 1 - 11 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + T + p T^{2} \) |
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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
−12.32903031423943753654199570723, −11.22514504460385312363737090829, −10.49034723509072962140764470290, −9.460094847454279126868063019693, −8.092105179993141599508863549634, −7.62368049380914397426496743518, −5.79729187121046804129336776899, −5.18432945826177342955590271236, −3.48503979598989868638885321849, −1.73582051333148490489274132763,
1.73582051333148490489274132763, 3.48503979598989868638885321849, 5.18432945826177342955590271236, 5.79729187121046804129336776899, 7.62368049380914397426496743518, 8.092105179993141599508863549634, 9.460094847454279126868063019693, 10.49034723509072962140764470290, 11.22514504460385312363737090829, 12.32903031423943753654199570723
