| L(s) = 1 | + (0.925 − 1.06i)2-s + (0.157 − 0.993i)3-s + (−0.286 − 1.97i)4-s + (−0.389 − 2.20i)5-s + (−0.916 − 1.08i)6-s + 0.222i·7-s + (−2.38 − 1.52i)8-s + (1.89 + 0.614i)9-s + (−2.71 − 1.62i)10-s + (0.721 − 1.41i)11-s + (−2.01 − 0.0271i)12-s + (1.79 + 3.51i)13-s + (0.237 + 0.205i)14-s + (−2.24 + 0.0406i)15-s + (−3.83 + 1.13i)16-s + (−1.68 − 1.22i)17-s + ⋯ |
| L(s) = 1 | + (0.654 − 0.755i)2-s + (0.0908 − 0.573i)3-s + (−0.143 − 0.989i)4-s + (−0.174 − 0.984i)5-s + (−0.374 − 0.444i)6-s + 0.0840i·7-s + (−0.841 − 0.539i)8-s + (0.630 + 0.204i)9-s + (−0.858 − 0.512i)10-s + (0.217 − 0.426i)11-s + (−0.580 − 0.00785i)12-s + (0.497 + 0.975i)13-s + (0.0635 + 0.0550i)14-s + (−0.580 + 0.0104i)15-s + (−0.959 + 0.283i)16-s + (−0.408 − 0.296i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 400 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.812 + 0.582i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.571392 - 1.77860i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.571392 - 1.77860i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.925 + 1.06i)T \) |
| 5 | \( 1 + (0.389 + 2.20i)T \) |
| good | 3 | \( 1 + (-0.157 + 0.993i)T + (-2.85 - 0.927i)T^{2} \) |
| 7 | \( 1 - 0.222iT - 7T^{2} \) |
| 11 | \( 1 + (-0.721 + 1.41i)T + (-6.46 - 8.89i)T^{2} \) |
| 13 | \( 1 + (-1.79 - 3.51i)T + (-7.64 + 10.5i)T^{2} \) |
| 17 | \( 1 + (1.68 + 1.22i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (7.18 - 1.13i)T + (18.0 - 5.87i)T^{2} \) |
| 23 | \( 1 + (-4.29 + 1.39i)T + (18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-0.298 + 1.88i)T + (-27.5 - 8.96i)T^{2} \) |
| 31 | \( 1 + (-7.10 - 5.16i)T + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (-6.20 + 3.16i)T + (21.7 - 29.9i)T^{2} \) |
| 41 | \( 1 + (-9.90 - 3.21i)T + (33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 + (8.51 + 8.51i)T + 43iT^{2} \) |
| 47 | \( 1 + (3.95 - 2.87i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (-1.59 - 0.252i)T + (50.4 + 16.3i)T^{2} \) |
| 59 | \( 1 + (-2.98 + 1.52i)T + (34.6 - 47.7i)T^{2} \) |
| 61 | \( 1 + (-8.16 - 4.15i)T + (35.8 + 49.3i)T^{2} \) |
| 67 | \( 1 + (6.76 - 1.07i)T + (63.7 - 20.7i)T^{2} \) |
| 71 | \( 1 + (1.31 + 1.81i)T + (-21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (5.28 - 1.71i)T + (59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (6.95 - 5.05i)T + (24.4 - 75.1i)T^{2} \) |
| 83 | \( 1 + (-9.17 + 1.45i)T + (78.9 - 25.6i)T^{2} \) |
| 89 | \( 1 + (13.0 - 4.25i)T + (72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-6.25 + 4.54i)T + (29.9 - 92.2i)T^{2} \) |
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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
−11.16808257047248015016673669567, −10.15783612804072828120839698172, −9.039303422214806018517452716242, −8.416335962193444071054861279182, −6.90301513450556978926612737609, −6.05477220310078944443982048315, −4.67704402514768594390329160629, −4.07059531569466651303640962832, −2.30571960048445214255387162230, −1.09849746745187984922509379996,
2.73269374338708406970933090257, 3.87967430438457278545163610515, 4.63133234036499585867084560324, 6.09681846473707659932112231649, 6.77929269311961918421605413199, 7.74892808699534777792016772100, 8.719988593346849399915862215655, 9.892070475529154019087062251614, 10.75034427874419702511783747417, 11.59337821095786733112500468686
