| L(s) = 1 | + 4i·2-s − 16·4-s − 141. i·7-s − 64i·8-s + 113.·11-s + 61.7i·13-s + 566.·14-s + 256·16-s + 1.67e3i·17-s + 662.·19-s + 453. i·22-s − 86.4i·23-s − 246.·26-s + 2.26e3i·28-s + 3.23e3·29-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s − 0.5·4-s − 1.09i·7-s − 0.353i·8-s + 0.282·11-s + 0.101i·13-s + 0.772·14-s + 0.250·16-s + 1.40i·17-s + 0.420·19-s + 0.199i·22-s − 0.0340i·23-s − 0.0716·26-s + 0.546i·28-s + 0.713·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (0.447 + 0.894i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(1.271294910\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.271294910\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 4iT \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 141. iT - 1.68e4T^{2} \) |
| 11 | \( 1 - 113.T + 1.61e5T^{2} \) |
| 13 | \( 1 - 61.7iT - 3.71e5T^{2} \) |
| 17 | \( 1 - 1.67e3iT - 1.41e6T^{2} \) |
| 19 | \( 1 - 662.T + 2.47e6T^{2} \) |
| 23 | \( 1 + 86.4iT - 6.43e6T^{2} \) |
| 29 | \( 1 - 3.23e3T + 2.05e7T^{2} \) |
| 31 | \( 1 + 3.81e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + 1.02e4iT - 6.93e7T^{2} \) |
| 41 | \( 1 + 1.34e4T + 1.15e8T^{2} \) |
| 43 | \( 1 + 4.69e3iT - 1.47e8T^{2} \) |
| 47 | \( 1 + 1.52e4iT - 2.29e8T^{2} \) |
| 53 | \( 1 + 498. iT - 4.18e8T^{2} \) |
| 59 | \( 1 - 1.52e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 3.18e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + 4.92e4iT - 1.35e9T^{2} \) |
| 71 | \( 1 - 2.10e4T + 1.80e9T^{2} \) |
| 73 | \( 1 - 3.94e4iT - 2.07e9T^{2} \) |
| 79 | \( 1 + 7.29e4T + 3.07e9T^{2} \) |
| 83 | \( 1 + 1.00e5iT - 3.93e9T^{2} \) |
| 89 | \( 1 - 1.46e5T + 5.58e9T^{2} \) |
| 97 | \( 1 - 4.35e4iT - 8.58e9T^{2} \) |
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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
−10.17391475162081779380342888214, −9.084328722660973743138505465201, −8.184272678577210434522349948788, −7.29217012673260272292628271143, −6.53319820470295956912767036950, −5.48336996101505174359639949362, −4.29292408410368509969550759474, −3.51513813819747420924158360374, −1.63366005478907847984130369464, −0.33403810327349186413945292587,
1.12478884627645462471873694238, 2.43210920727522570763151558566, 3.26937469156100355820515316522, 4.70007357338820478692873572986, 5.50756433103629154047430896741, 6.68375022076123948206592275496, 7.904649151375885902559764892527, 8.913802669358788130039920045114, 9.502936503533354224204489425796, 10.44398945282374269392983238209
