| L(s) = 1 | − 16.7·3-s + 25·5-s − 94.1·7-s + 36.4·9-s − 143.·11-s + 421.·13-s − 417.·15-s + 1.98e3·17-s + 1.31e3·19-s + 1.57e3·21-s + 4.02e3·23-s + 625·25-s + 3.45e3·27-s − 6.41e3·29-s + 2.35e3·31-s + 2.40e3·33-s − 2.35e3·35-s − 7.87e3·37-s − 7.04e3·39-s + 1.50e4·41-s − 1.14e3·43-s + 910.·45-s + 2.15e4·47-s − 7.94e3·49-s − 3.31e4·51-s + 9.56e3·53-s − 3.59e3·55-s + ⋯ |
| L(s) = 1 | − 1.07·3-s + 0.447·5-s − 0.726·7-s + 0.149·9-s − 0.358·11-s + 0.691·13-s − 0.479·15-s + 1.66·17-s + 0.837·19-s + 0.778·21-s + 1.58·23-s + 0.200·25-s + 0.911·27-s − 1.41·29-s + 0.439·31-s + 0.383·33-s − 0.324·35-s − 0.945·37-s − 0.741·39-s + 1.40·41-s − 0.0941·43-s + 0.0670·45-s + 1.42·47-s − 0.472·49-s − 1.78·51-s + 0.467·53-s − 0.160·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.187777183\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.187777183\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 5 | \( 1 - 25T \) |
| good | 3 | \( 1 + 16.7T + 243T^{2} \) |
| 7 | \( 1 + 94.1T + 1.68e4T^{2} \) |
| 11 | \( 1 + 143.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 421.T + 3.71e5T^{2} \) |
| 17 | \( 1 - 1.98e3T + 1.41e6T^{2} \) |
| 19 | \( 1 - 1.31e3T + 2.47e6T^{2} \) |
| 23 | \( 1 - 4.02e3T + 6.43e6T^{2} \) |
| 29 | \( 1 + 6.41e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.35e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 7.87e3T + 6.93e7T^{2} \) |
| 41 | \( 1 - 1.50e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 1.14e3T + 1.47e8T^{2} \) |
| 47 | \( 1 - 2.15e4T + 2.29e8T^{2} \) |
| 53 | \( 1 - 9.56e3T + 4.18e8T^{2} \) |
| 59 | \( 1 + 4.27e4T + 7.14e8T^{2} \) |
| 61 | \( 1 - 3.21e4T + 8.44e8T^{2} \) |
| 67 | \( 1 - 3.03e4T + 1.35e9T^{2} \) |
| 71 | \( 1 + 3.60e4T + 1.80e9T^{2} \) |
| 73 | \( 1 + 6.34e4T + 2.07e9T^{2} \) |
| 79 | \( 1 - 8.99e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 3.82e4T + 3.93e9T^{2} \) |
| 89 | \( 1 - 5.74e3T + 5.58e9T^{2} \) |
| 97 | \( 1 - 1.78e5T + 8.58e9T^{2} \) |
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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
−13.22141430387596731804573087869, −12.30307350352262584450903273939, −11.19926942396042857658251164258, −10.24098021989322024302160726647, −9.100889102275110912313548705786, −7.38097935374464368211514605799, −6.03448109439684576404105985846, −5.27784775488046631804627170362, −3.22327427245931833527234206242, −0.894891540791953808883245827389,
0.894891540791953808883245827389, 3.22327427245931833527234206242, 5.27784775488046631804627170362, 6.03448109439684576404105985846, 7.38097935374464368211514605799, 9.100889102275110912313548705786, 10.24098021989322024302160726647, 11.19926942396042857658251164258, 12.30307350352262584450903273939, 13.22141430387596731804573087869
