| L(s) = 1 | + (0.490 − 2.30i)2-s + (−2.00 − 0.211i)3-s + (−3.24 − 1.44i)4-s + (0.0220 − 0.0198i)5-s + (−1.47 + 4.52i)6-s + (2.06 − 1.65i)7-s + (−2.15 + 2.96i)8-s + (1.05 + 0.223i)9-s + (−0.0350 − 0.0606i)10-s + (2.86 − 1.67i)11-s + (6.21 + 3.58i)12-s + (1.27 + 3.91i)13-s + (−2.80 − 5.57i)14-s + (−0.0485 + 0.0352i)15-s + (1.02 + 1.13i)16-s + (3.81 − 0.811i)17-s + ⋯ |
| L(s) = 1 | + (0.346 − 1.63i)2-s + (−1.15 − 0.121i)3-s + (−1.62 − 0.723i)4-s + (0.00987 − 0.00889i)5-s + (−0.600 + 1.84i)6-s + (0.780 − 0.625i)7-s + (−0.761 + 1.04i)8-s + (0.351 + 0.0746i)9-s + (−0.0110 − 0.0191i)10-s + (0.862 − 0.505i)11-s + (1.79 + 1.03i)12-s + (0.352 + 1.08i)13-s + (−0.749 − 1.48i)14-s + (−0.0125 + 0.00910i)15-s + (0.255 + 0.284i)16-s + (0.925 − 0.196i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.838 + 0.545i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.228729 - 0.770604i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.228729 - 0.770604i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (-2.06 + 1.65i)T \) |
| 11 | \( 1 + (-2.86 + 1.67i)T \) |
| good | 2 | \( 1 + (-0.490 + 2.30i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (2.00 + 0.211i)T + (2.93 + 0.623i)T^{2} \) |
| 5 | \( 1 + (-0.0220 + 0.0198i)T + (0.522 - 4.97i)T^{2} \) |
| 13 | \( 1 + (-1.27 - 3.91i)T + (-10.5 + 7.64i)T^{2} \) |
| 17 | \( 1 + (-3.81 + 0.811i)T + (15.5 - 6.91i)T^{2} \) |
| 19 | \( 1 + (5.48 - 2.44i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-1.70 + 2.96i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-2.60 - 3.58i)T + (-8.96 + 27.5i)T^{2} \) |
| 31 | \( 1 + (2.55 + 2.29i)T + (3.24 + 30.8i)T^{2} \) |
| 37 | \( 1 + (-1.12 - 10.6i)T + (-36.1 + 7.69i)T^{2} \) |
| 41 | \( 1 + (4.39 + 3.19i)T + (12.6 + 38.9i)T^{2} \) |
| 43 | \( 1 + 4.40iT - 43T^{2} \) |
| 47 | \( 1 + (0.812 + 1.82i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-1.11 + 1.23i)T + (-5.54 - 52.7i)T^{2} \) |
| 59 | \( 1 + (-1.62 + 3.64i)T + (-39.4 - 43.8i)T^{2} \) |
| 61 | \( 1 + (5.78 + 6.42i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (-1.46 - 2.53i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (0.813 - 2.50i)T + (-57.4 - 41.7i)T^{2} \) |
| 73 | \( 1 + (-1.77 - 0.790i)T + (48.8 + 54.2i)T^{2} \) |
| 79 | \( 1 + (2.31 - 10.8i)T + (-72.1 - 32.1i)T^{2} \) |
| 83 | \( 1 + (2.78 - 8.56i)T + (-67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-1.57 - 0.909i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-10.7 + 3.49i)T + (78.4 - 57.0i)T^{2} \) |
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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
−13.77844952671355940899275645739, −12.51387515573490863043631022716, −11.62770468342698507109408446297, −11.13641110538162527661388636966, −10.20069943386025719655165732176, −8.728417661888998233907329240299, −6.63617846941332158128401137657, −5.05403146656059753969381100381, −3.82106109855860425596921638436, −1.37242723985141860512818925126,
4.51932247375347267966325877319, 5.58749829372322482020585089251, 6.34814904090240903742704360979, 7.76648148642146010867867635381, 8.857969492666059523510677385222, 10.61694741240260866005981302687, 11.84139333983015452543665676877, 12.86405788818015241416588784683, 14.33762226092990206479076169122, 15.02294079554590944432489792268
