| L(s) = 1 | + (−0.415 + 0.909i)2-s + (3.20 + 0.939i)3-s + (−0.654 − 0.755i)4-s + (−3.28 + 2.10i)5-s + (−2.18 + 2.52i)6-s + (−0.142 + 0.989i)7-s + (0.959 − 0.281i)8-s + (6.83 + 4.39i)9-s + (−0.554 − 3.85i)10-s + (−1.52 − 3.34i)11-s + (−1.38 − 3.03i)12-s + (0.481 + 3.34i)13-s + (−0.841 − 0.540i)14-s + (−12.4 + 3.66i)15-s + (−0.142 + 0.989i)16-s + (−0.142 + 0.164i)17-s + ⋯ |
| L(s) = 1 | + (−0.293 + 0.643i)2-s + (1.84 + 0.542i)3-s + (−0.327 − 0.377i)4-s + (−1.46 + 0.942i)5-s + (−0.891 + 1.02i)6-s + (−0.0537 + 0.374i)7-s + (0.339 − 0.0996i)8-s + (2.27 + 1.46i)9-s + (−0.175 − 1.22i)10-s + (−0.460 − 1.00i)11-s + (−0.400 − 0.875i)12-s + (0.133 + 0.928i)13-s + (−0.224 − 0.144i)14-s + (−3.22 + 0.946i)15-s + (−0.0355 + 0.247i)16-s + (−0.0346 + 0.0400i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 322 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.468 - 0.883i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.803590 + 1.33634i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.803590 + 1.33634i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.415 - 0.909i)T \) |
| 7 | \( 1 + (0.142 - 0.989i)T \) |
| 23 | \( 1 + (2.14 + 4.29i)T \) |
| good | 3 | \( 1 + (-3.20 - 0.939i)T + (2.52 + 1.62i)T^{2} \) |
| 5 | \( 1 + (3.28 - 2.10i)T + (2.07 - 4.54i)T^{2} \) |
| 11 | \( 1 + (1.52 + 3.34i)T + (-7.20 + 8.31i)T^{2} \) |
| 13 | \( 1 + (-0.481 - 3.34i)T + (-12.4 + 3.66i)T^{2} \) |
| 17 | \( 1 + (0.142 - 0.164i)T + (-2.41 - 16.8i)T^{2} \) |
| 19 | \( 1 + (-2.07 - 2.39i)T + (-2.70 + 18.8i)T^{2} \) |
| 29 | \( 1 + (-2.71 + 3.13i)T + (-4.12 - 28.7i)T^{2} \) |
| 31 | \( 1 + (-2.23 + 0.657i)T + (26.0 - 16.7i)T^{2} \) |
| 37 | \( 1 + (-5.22 - 3.35i)T + (15.3 + 33.6i)T^{2} \) |
| 41 | \( 1 + (-5.39 + 3.46i)T + (17.0 - 37.2i)T^{2} \) |
| 43 | \( 1 + (-2.00 - 0.589i)T + (36.1 + 23.2i)T^{2} \) |
| 47 | \( 1 + 6.78T + 47T^{2} \) |
| 53 | \( 1 + (-0.803 + 5.59i)T + (-50.8 - 14.9i)T^{2} \) |
| 59 | \( 1 + (0.0301 + 0.209i)T + (-56.6 + 16.6i)T^{2} \) |
| 61 | \( 1 + (-2.40 + 0.705i)T + (51.3 - 32.9i)T^{2} \) |
| 67 | \( 1 + (-2.10 + 4.59i)T + (-43.8 - 50.6i)T^{2} \) |
| 71 | \( 1 + (3.35 - 7.33i)T + (-46.4 - 53.6i)T^{2} \) |
| 73 | \( 1 + (5.08 + 5.86i)T + (-10.3 + 72.2i)T^{2} \) |
| 79 | \( 1 + (0.900 + 6.25i)T + (-75.7 + 22.2i)T^{2} \) |
| 83 | \( 1 + (10.9 + 7.06i)T + (34.4 + 75.4i)T^{2} \) |
| 89 | \( 1 + (-17.9 - 5.26i)T + (74.8 + 48.1i)T^{2} \) |
| 97 | \( 1 + (12.1 - 7.78i)T + (40.2 - 88.2i)T^{2} \) |
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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
−11.78690796640249309589286563001, −10.73395384419591563989881070904, −9.852936428282965838370908575058, −8.771933801584602657753836655550, −8.137351636690198678995310847301, −7.59953745319412788368203804727, −6.46431259649990643035088761188, −4.48786412306053936532987411052, −3.62238695912712099119145244372, −2.62816687026864070574784617954,
1.13897168386410130607811500003, 2.78588290528156680376311033621, 3.77868961521482038358404439657, 4.66231659773507400953902980450, 7.30470042923764952159486746859, 7.72399839166244833099716704737, 8.416485568879030662070283316199, 9.270716746109143988461542919436, 10.10784791043040687014275731114, 11.50654038660858049492337972782
