| L(s) = 1 | − 9.22·3-s − 25·5-s − 49·7-s − 157.·9-s − 220.·11-s + 928.·13-s + 230.·15-s − 1.92e3·17-s + 1.92e3·19-s + 452.·21-s − 1.61e3·23-s + 625·25-s + 3.69e3·27-s + 4.66e3·29-s + 5.19e3·31-s + 2.03e3·33-s + 1.22e3·35-s + 1.29e4·37-s − 8.56e3·39-s − 6.44e3·41-s + 2.91e3·43-s + 3.94e3·45-s + 2.36e4·47-s + 2.40e3·49-s + 1.77e4·51-s + 2.78e4·53-s + 5.52e3·55-s + ⋯ |
| L(s) = 1 | − 0.591·3-s − 0.447·5-s − 0.377·7-s − 0.649·9-s − 0.550·11-s + 1.52·13-s + 0.264·15-s − 1.61·17-s + 1.22·19-s + 0.223·21-s − 0.637·23-s + 0.200·25-s + 0.976·27-s + 1.02·29-s + 0.971·31-s + 0.325·33-s + 0.169·35-s + 1.55·37-s − 0.901·39-s − 0.599·41-s + 0.240·43-s + 0.290·45-s + 1.55·47-s + 0.142·49-s + 0.955·51-s + 1.36·53-s + 0.246·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.087047062\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.087047062\) |
| \(L(\frac{7}{2})\) |
|
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 | \( 1 + 9.22T + 243T^{2} \) |
| 11 | \( 1 + 220.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 928.T + 3.71e5T^{2} \) |
| 17 | \( 1 + 1.92e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.92e3T + 2.47e6T^{2} \) |
| 23 | \( 1 + 1.61e3T + 6.43e6T^{2} \) |
| 29 | \( 1 - 4.66e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 5.19e3T + 2.86e7T^{2} \) |
| 37 | \( 1 - 1.29e4T + 6.93e7T^{2} \) |
| 41 | \( 1 + 6.44e3T + 1.15e8T^{2} \) |
| 43 | \( 1 - 2.91e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.36e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 2.78e4T + 4.18e8T^{2} \) |
| 59 | \( 1 + 8.71e3T + 7.14e8T^{2} \) |
| 61 | \( 1 + 5.24e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 2.71e4T + 1.35e9T^{2} \) |
| 71 | \( 1 - 6.05e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 4.42e4T + 2.07e9T^{2} \) |
| 79 | \( 1 + 5.46e3T + 3.07e9T^{2} \) |
| 83 | \( 1 + 8.68e4T + 3.93e9T^{2} \) |
| 89 | \( 1 + 3.30e4T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.45e5T + 8.58e9T^{2} \) |
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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
−12.02406677720653813954678660211, −11.27429145009260907674606956019, −10.45057004213452527838268603052, −9.003122113386943959989371401155, −8.079717058948493485048285771873, −6.62692103584948391851283198429, −5.72639155407656736267830431238, −4.30655154044068854156851729064, −2.83764528637882106458836264564, −0.70723217144732657831299376202,
0.70723217144732657831299376202, 2.83764528637882106458836264564, 4.30655154044068854156851729064, 5.72639155407656736267830431238, 6.62692103584948391851283198429, 8.079717058948493485048285771873, 9.003122113386943959989371401155, 10.45057004213452527838268603052, 11.27429145009260907674606956019, 12.02406677720653813954678660211
