| L(s) = 1 | − 3-s + 4·7-s + 9-s + 4·11-s + 2·13-s + 6·17-s − 4·19-s − 4·21-s − 27-s + 2·29-s + 4·31-s − 4·33-s + 2·37-s − 2·39-s + 2·41-s − 4·43-s − 8·47-s + 9·49-s − 6·51-s − 10·53-s + 4·57-s − 4·59-s + 6·61-s + 4·63-s − 4·67-s − 16·71-s + 6·73-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1.51·7-s + 1/3·9-s + 1.20·11-s + 0.554·13-s + 1.45·17-s − 0.917·19-s − 0.872·21-s − 0.192·27-s + 0.371·29-s + 0.718·31-s − 0.696·33-s + 0.328·37-s − 0.320·39-s + 0.312·41-s − 0.609·43-s − 1.16·47-s + 9/7·49-s − 0.840·51-s − 1.37·53-s + 0.529·57-s − 0.520·59-s + 0.768·61-s + 0.503·63-s − 0.488·67-s − 1.89·71-s + 0.702·73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.095499950\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.095499950\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| good | 7 | \( 1 - 4 T + p T^{2} \) |
| 11 | \( 1 - 4 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 6 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 - 2 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 - 2 T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + 8 T + p T^{2} \) |
| 53 | \( 1 + 10 T + p T^{2} \) |
| 59 | \( 1 + 4 T + p T^{2} \) |
| 61 | \( 1 - 6 T + p T^{2} \) |
| 67 | \( 1 + 4 T + p T^{2} \) |
| 71 | \( 1 + 16 T + p T^{2} \) |
| 73 | \( 1 - 6 T + p T^{2} \) |
| 79 | \( 1 - 4 T + p T^{2} \) |
| 83 | \( 1 + 12 T + p T^{2} \) |
| 89 | \( 1 - 10 T + p T^{2} \) |
| 97 | \( 1 - 14 T + p T^{2} \) |
| show more | |
| show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−8.824046048556980758736864363820, −8.187690885601832957542462060811, −7.51599841504707801181092960196, −6.48884074088343388628451242584, −5.91022112565038900100369808945, −4.90911980616858985140087544886, −4.34933392140332560567015192416, −3.33595145158661942591347021628, −1.79509095105059034627127171893, −1.07872352480771079736145349824,
1.07872352480771079736145349824, 1.79509095105059034627127171893, 3.33595145158661942591347021628, 4.34933392140332560567015192416, 4.90911980616858985140087544886, 5.91022112565038900100369808945, 6.48884074088343388628451242584, 7.51599841504707801181092960196, 8.187690885601832957542462060811, 8.824046048556980758736864363820
