| 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
−15.02294079554590944432489792268, −14.33762226092990206479076169122, −12.86405788818015241416588784683, −11.84139333983015452543665676877, −10.61694741240260866005981302687, −8.857969492666059523510677385222, −7.76648148642146010867867635381, −6.34814904090240903742704360979, −5.58749829372322482020585089251, −4.51932247375347267966325877319,
1.37242723985141860512818925126, 3.82106109855860425596921638436, 5.05403146656059753969381100381, 6.63617846941332158128401137657, 8.728417661888998233907329240299, 10.20069943386025719655165732176, 11.13641110538162527661388636966, 11.62770468342698507109408446297, 12.51387515573490863043631022716, 13.77844952671355940899275645739
