| L(s) = 1 | − 2·2-s + 4·4-s + 5i·5-s − 6.26·7-s − 8·8-s − 10i·10-s − 46.2i·11-s − 34.4i·13-s + 12.5·14-s + 16·16-s − 13.4i·17-s + (−76.7 + 31.1i)19-s + 20i·20-s + 92.5i·22-s + 111. i·23-s + ⋯ |
| L(s) = 1 | − 0.707·2-s + 0.5·4-s + 0.447i·5-s − 0.338·7-s − 0.353·8-s − 0.316i·10-s − 1.26i·11-s − 0.734i·13-s + 0.239·14-s + 0.250·16-s − 0.191i·17-s + (−0.926 + 0.376i)19-s + 0.223i·20-s + 0.897i·22-s + 1.01i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(4-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1710 ^{s/2} \, \Gamma_{\C}(s+3/2) \, L(s)\cr =\mathstrut & (0.539 - 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(2)\) |
\(\approx\) |
\(0.8525807448\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.8525807448\) |
| \(L(\frac{5}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + 2T \) |
| 3 | \( 1 \) |
| 5 | \( 1 - 5iT \) |
| 19 | \( 1 + (76.7 - 31.1i)T \) |
| good | 7 | \( 1 + 6.26T + 343T^{2} \) |
| 11 | \( 1 + 46.2iT - 1.33e3T^{2} \) |
| 13 | \( 1 + 34.4iT - 2.19e3T^{2} \) |
| 17 | \( 1 + 13.4iT - 4.91e3T^{2} \) |
| 23 | \( 1 - 111. iT - 1.21e4T^{2} \) |
| 29 | \( 1 + 144.T + 2.43e4T^{2} \) |
| 31 | \( 1 - 30.8iT - 2.97e4T^{2} \) |
| 37 | \( 1 - 60.7iT - 5.06e4T^{2} \) |
| 41 | \( 1 + 39.9T + 6.89e4T^{2} \) |
| 43 | \( 1 + 422.T + 7.95e4T^{2} \) |
| 47 | \( 1 - 33.6iT - 1.03e5T^{2} \) |
| 53 | \( 1 - 600.T + 1.48e5T^{2} \) |
| 59 | \( 1 - 711.T + 2.05e5T^{2} \) |
| 61 | \( 1 - 576.T + 2.26e5T^{2} \) |
| 67 | \( 1 + 269. iT - 3.00e5T^{2} \) |
| 71 | \( 1 - 291.T + 3.57e5T^{2} \) |
| 73 | \( 1 + 457.T + 3.89e5T^{2} \) |
| 79 | \( 1 + 949. iT - 4.93e5T^{2} \) |
| 83 | \( 1 + 737. iT - 5.71e5T^{2} \) |
| 89 | \( 1 + 1.47e3T + 7.04e5T^{2} \) |
| 97 | \( 1 - 469. iT - 9.12e5T^{2} \) |
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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
−8.986833515616664883110260139765, −8.374514094874114973066726999969, −7.63526707413492644172943264370, −6.76066584267595508503610603990, −6.00665150253677384426162211796, −5.28316919368172659285429316303, −3.71981976122570079283215732171, −3.12943608323056128199596864196, −1.98168638334851472292297528894, −0.69027769602490276458814866920,
0.33329060154657019840051902309, 1.71520331111995825841646854930, 2.42578449844618102609404370672, 3.87971901153529984150078397689, 4.64388888162492827701161202188, 5.69332691162347213565142193687, 6.82547244490861091120663290342, 7.07089314226408665330132144064, 8.348215487137121288504405112866, 8.724610160780983272272962537714
