| L(s) = 1 | + 3·5-s − 7-s − 3·11-s + 2·13-s − 6·17-s − 5·19-s + 9·23-s + 4·25-s + 6·29-s + 31-s − 3·35-s + 11·37-s + 3·41-s + 4·43-s + 12·47-s + 49-s − 9·55-s + 8·61-s + 6·65-s + 10·67-s − 3·71-s + 8·73-s + 3·77-s + 4·79-s − 6·83-s − 18·85-s + 3·89-s + ⋯ |
| L(s) = 1 | + 1.34·5-s − 0.377·7-s − 0.904·11-s + 0.554·13-s − 1.45·17-s − 1.14·19-s + 1.87·23-s + 4/5·25-s + 1.11·29-s + 0.179·31-s − 0.507·35-s + 1.80·37-s + 0.468·41-s + 0.609·43-s + 1.75·47-s + 1/7·49-s − 1.21·55-s + 1.02·61-s + 0.744·65-s + 1.22·67-s − 0.356·71-s + 0.936·73-s + 0.341·77-s + 0.450·79-s − 0.658·83-s − 1.95·85-s + 0.317·89-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3024 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.189143294\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.189143294\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 + T \) |
| good | 5 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 3 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 + 6 T + p T^{2} \) |
| 19 | \( 1 + 5 T + p T^{2} \) |
| 23 | \( 1 - 9 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 - 11 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 12 T + p T^{2} \) |
| 53 | \( 1 + p T^{2} \) |
| 59 | \( 1 + p T^{2} \) |
| 61 | \( 1 - 8 T + p T^{2} \) |
| 67 | \( 1 - 10 T + p T^{2} \) |
| 71 | \( 1 + 3 T + p T^{2} \) |
| 73 | \( 1 - 8 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 3 T + p T^{2} \) |
| 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
−8.889292709358640155902653837334, −8.117026505983201952711362349952, −6.97788966272376598896013488982, −6.42431345949670830286643881465, −5.77456565940329463172933674847, −4.93205334497940119535416883331, −4.11606738245412655571366680184, −2.67436548689122413437384246797, −2.34452675669588835298859263834, −0.906362563217599442528963783994,
0.906362563217599442528963783994, 2.34452675669588835298859263834, 2.67436548689122413437384246797, 4.11606738245412655571366680184, 4.93205334497940119535416883331, 5.77456565940329463172933674847, 6.42431345949670830286643881465, 6.97788966272376598896013488982, 8.117026505983201952711362349952, 8.889292709358640155902653837334
