| L(s) = 1 | − 8·2-s + 2.39·3-s + 64·4-s − 162.·5-s − 19.1·6-s + 677.·7-s − 512·8-s − 2.18e3·9-s + 1.29e3·10-s − 5.83e3·11-s + 153.·12-s + 7.76e3·13-s − 5.42e3·14-s − 388.·15-s + 4.09e3·16-s + 3.05e4·17-s + 1.74e4·18-s − 2.00e4·19-s − 1.03e4·20-s + 1.62e3·21-s + 4.67e4·22-s − 2.06e4·23-s − 1.22e3·24-s − 5.18e4·25-s − 6.21e4·26-s − 1.04e4·27-s + 4.33e4·28-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.0512·3-s + 0.5·4-s − 0.580·5-s − 0.0362·6-s + 0.746·7-s − 0.353·8-s − 0.997·9-s + 0.410·10-s − 1.32·11-s + 0.0256·12-s + 0.980·13-s − 0.528·14-s − 0.0297·15-s + 0.250·16-s + 1.50·17-s + 0.705·18-s − 0.670·19-s − 0.290·20-s + 0.0382·21-s + 0.935·22-s − 0.354·23-s − 0.0181·24-s − 0.663·25-s − 0.693·26-s − 0.102·27-s + 0.373·28-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(8-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 538 ^{s/2} \, \Gamma_{\C}(s+7/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(4)\) |
\(\approx\) |
\(0.8818037000\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8818037000\) |
| \(L(\frac{9}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 8T \) |
| 269 | \( 1 + 1.94e7T \) |
| good | 3 | \( 1 - 2.39T + 2.18e3T^{2} \) |
| 5 | \( 1 + 162.T + 7.81e4T^{2} \) |
| 7 | \( 1 - 677.T + 8.23e5T^{2} \) |
| 11 | \( 1 + 5.83e3T + 1.94e7T^{2} \) |
| 13 | \( 1 - 7.76e3T + 6.27e7T^{2} \) |
| 17 | \( 1 - 3.05e4T + 4.10e8T^{2} \) |
| 19 | \( 1 + 2.00e4T + 8.93e8T^{2} \) |
| 23 | \( 1 + 2.06e4T + 3.40e9T^{2} \) |
| 29 | \( 1 - 6.42e4T + 1.72e10T^{2} \) |
| 31 | \( 1 + 3.01e5T + 2.75e10T^{2} \) |
| 37 | \( 1 - 1.49e5T + 9.49e10T^{2} \) |
| 41 | \( 1 + 1.98e5T + 1.94e11T^{2} \) |
| 43 | \( 1 - 1.41e5T + 2.71e11T^{2} \) |
| 47 | \( 1 - 7.38e5T + 5.06e11T^{2} \) |
| 53 | \( 1 + 6.58e5T + 1.17e12T^{2} \) |
| 59 | \( 1 + 3.05e5T + 2.48e12T^{2} \) |
| 61 | \( 1 + 1.52e5T + 3.14e12T^{2} \) |
| 67 | \( 1 - 3.75e6T + 6.06e12T^{2} \) |
| 71 | \( 1 - 2.85e6T + 9.09e12T^{2} \) |
| 73 | \( 1 + 4.51e6T + 1.10e13T^{2} \) |
| 79 | \( 1 + 3.43e4T + 1.92e13T^{2} \) |
| 83 | \( 1 + 8.98e6T + 2.71e13T^{2} \) |
| 89 | \( 1 - 2.09e6T + 4.42e13T^{2} \) |
| 97 | \( 1 + 1.21e7T + 8.07e13T^{2} \) |
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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
−9.664496878214983444223585741957, −8.468544916315708045165602945354, −8.115123954848724560883015174236, −7.39682058776330682510369912118, −5.95295961416069673424933696136, −5.29865121189650288668412628038, −3.84797018895690369928062816468, −2.83663677001601010039740271723, −1.69065033380868536861233220455, −0.44728684271588943537623775702,
0.44728684271588943537623775702, 1.69065033380868536861233220455, 2.83663677001601010039740271723, 3.84797018895690369928062816468, 5.29865121189650288668412628038, 5.95295961416069673424933696136, 7.39682058776330682510369912118, 8.115123954848724560883015174236, 8.468544916315708045165602945354, 9.664496878214983444223585741957
