| L(s) = 1 | + 16.5·3-s + 25·5-s + 49·7-s + 29.3·9-s − 680.·11-s + 866.·13-s + 412.·15-s + 606.·17-s + 1.62e3·19-s + 808.·21-s + 2.28e3·23-s + 625·25-s − 3.52e3·27-s + 8.76e3·29-s + 5.33e3·31-s − 1.12e4·33-s + 1.22e3·35-s − 1.19e4·37-s + 1.42e4·39-s + 2.91e3·41-s + 7.42e3·43-s + 733.·45-s + 2.77e3·47-s + 2.40e3·49-s + 1.00e4·51-s + 3.12e4·53-s − 1.70e4·55-s + ⋯ |
| L(s) = 1 | + 1.05·3-s + 0.447·5-s + 0.377·7-s + 0.120·9-s − 1.69·11-s + 1.42·13-s + 0.473·15-s + 0.508·17-s + 1.02·19-s + 0.400·21-s + 0.899·23-s + 0.200·25-s − 0.930·27-s + 1.93·29-s + 0.997·31-s − 1.79·33-s + 0.169·35-s − 1.43·37-s + 1.50·39-s + 0.270·41-s + 0.612·43-s + 0.0539·45-s + 0.183·47-s + 0.142·49-s + 0.538·51-s + 1.52·53-s − 0.758·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(3.470288362\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.470288362\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| 7 | \( 1 - 49T \) |
| good | 3 | \( 1 - 16.5T + 243T^{2} \) |
| 11 | \( 1 + 680.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 866.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 606.T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.62e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 2.28e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 8.76e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.33e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.19e4T + 6.93e7T^{2} \) |
| 41 | \( 1 - 2.91e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 7.42e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.77e3T + 2.29e8T^{2} \) |
| 53 | \( 1 - 3.12e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 1.22e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 2.72e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 1.50e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.62e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 5.05e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 4.94e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 4.64e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 8.47e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 5.98e4T + 8.58e9T^{2} \) |
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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
−10.78439044848118710350927587069, −10.09222909315039203841589262664, −8.851800902290353821228799898923, −8.287990348860258182130815298560, −7.38336815099040317643663128315, −5.91477804335380785657986465922, −4.92965258894169621162018710222, −3.31567293072768676370663262547, −2.54963616492546237563332231851, −1.07977122861935902003253480741,
1.07977122861935902003253480741, 2.54963616492546237563332231851, 3.31567293072768676370663262547, 4.92965258894169621162018710222, 5.91477804335380785657986465922, 7.38336815099040317643663128315, 8.287990348860258182130815298560, 8.851800902290353821228799898923, 10.09222909315039203841589262664, 10.78439044848118710350927587069
