| L(s) = 1 | + 3-s − 5·7-s + 9-s + 5·11-s + 13-s + 5·17-s + 3·19-s − 5·21-s + 3·23-s + 27-s + 6·29-s + 6·31-s + 5·33-s + 37-s + 39-s + 4·43-s + 18·49-s + 5·51-s + 3·53-s + 3·57-s + 10·59-s + 10·61-s − 5·63-s − 14·67-s + 3·69-s − 6·71-s + 9·73-s + ⋯ |
| L(s) = 1 | + 0.577·3-s − 1.88·7-s + 1/3·9-s + 1.50·11-s + 0.277·13-s + 1.21·17-s + 0.688·19-s − 1.09·21-s + 0.625·23-s + 0.192·27-s + 1.11·29-s + 1.07·31-s + 0.870·33-s + 0.164·37-s + 0.160·39-s + 0.609·43-s + 18/7·49-s + 0.700·51-s + 0.412·53-s + 0.397·57-s + 1.30·59-s + 1.28·61-s − 0.629·63-s − 1.71·67-s + 0.361·69-s − 0.712·71-s + 1.05·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 44400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.576523397\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.576523397\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 - T \) |
| 5 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 7 | \( 1 + 5 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 - 5 T + p T^{2} \) |
| 19 | \( 1 - 3 T + p T^{2} \) |
| 23 | \( 1 - 3 T + p T^{2} \) |
| 29 | \( 1 - 6 T + p T^{2} \) |
| 31 | \( 1 - 6 T + p T^{2} \) |
| 41 | \( 1 + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 - 10 T + p T^{2} \) |
| 67 | \( 1 + 14 T + p T^{2} \) |
| 71 | \( 1 + 6 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + p T^{2} \) |
| 83 | \( 1 - 5 T + p T^{2} \) |
| 89 | \( 1 - 13 T + p T^{2} \) |
| 97 | \( 1 - 4 T + p T^{2} \) |
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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
−14.64889412485269, −14.11486839427711, −13.56137963722350, −13.28765028311559, −12.54667738537121, −12.09097194411312, −11.84527517192751, −10.98073386482257, −10.14123115312396, −9.953697492219796, −9.431376664362101, −8.938221782640028, −8.491525432970347, −7.678374834387223, −7.057902483671140, −6.645360164564924, −6.141692674498877, −5.595955126291237, −4.671760394542480, −3.925666513532790, −3.463564359412495, −3.032930605674306, −2.368827739249123, −1.142019719843789, −0.7911815961988398,
0.7911815961988398, 1.142019719843789, 2.368827739249123, 3.032930605674306, 3.463564359412495, 3.925666513532790, 4.671760394542480, 5.595955126291237, 6.141692674498877, 6.645360164564924, 7.057902483671140, 7.678374834387223, 8.491525432970347, 8.938221782640028, 9.431376664362101, 9.953697492219796, 10.14123115312396, 10.98073386482257, 11.84527517192751, 12.09097194411312, 12.54667738537121, 13.28765028311559, 13.56137963722350, 14.11486839427711, 14.64889412485269
