| L(s) = 1 | − 5-s − 7-s − 3·9-s + 6·11-s + 6·13-s + 7·17-s + 2·19-s − 23-s + 25-s − 5·29-s + 31-s + 35-s − 5·37-s − 7·41-s + 8·43-s + 3·45-s + 8·47-s − 6·49-s + 3·53-s − 6·55-s + 13·59-s − 8·61-s + 3·63-s − 6·65-s − 9·67-s + 7·71-s − 2·73-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.377·7-s − 9-s + 1.80·11-s + 1.66·13-s + 1.69·17-s + 0.458·19-s − 0.208·23-s + 1/5·25-s − 0.928·29-s + 0.179·31-s + 0.169·35-s − 0.821·37-s − 1.09·41-s + 1.21·43-s + 0.447·45-s + 1.16·47-s − 6/7·49-s + 0.412·53-s − 0.809·55-s + 1.69·59-s − 1.02·61-s + 0.377·63-s − 0.744·65-s − 1.09·67-s + 0.830·71-s − 0.234·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 460 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.385469857\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.385469857\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 + p T^{2} \) |
| 7 | \( 1 + T + p T^{2} \) |
| 11 | \( 1 - 6 T + p T^{2} \) |
| 13 | \( 1 - 6 T + p T^{2} \) |
| 17 | \( 1 - 7 T + p T^{2} \) |
| 19 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 5 T + p T^{2} \) |
| 31 | \( 1 - T + p T^{2} \) |
| 37 | \( 1 + 5 T + p T^{2} \) |
| 41 | \( 1 + 7 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 - 8 T + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 13 T + p T^{2} \) |
| 61 | \( 1 + 8 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 - 7 T + p T^{2} \) |
| 73 | \( 1 + 2 T + p T^{2} \) |
| 79 | \( 1 + 12 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 12 T + p T^{2} \) |
| 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
−11.32293125895052523712070840933, −10.15665611896575079666929529115, −9.080683710496085583783063281158, −8.528078535604299289260642773668, −7.39317105219739402615535339322, −6.27424663200910569544974372979, −5.59466245708033599078037736922, −3.88092971104947044377992240121, −3.32299209395225585747866958895, −1.22587440112496978750915360361,
1.22587440112496978750915360361, 3.32299209395225585747866958895, 3.88092971104947044377992240121, 5.59466245708033599078037736922, 6.27424663200910569544974372979, 7.39317105219739402615535339322, 8.528078535604299289260642773668, 9.080683710496085583783063281158, 10.15665611896575079666929529115, 11.32293125895052523712070840933
