| L(s) = 1 | + (−1.78 + 0.393i)2-s + (−1.26 − 1.18i)3-s + (1.22 − 0.567i)4-s + (1.62 + 0.452i)5-s + (2.72 + 1.62i)6-s + (−1.02 + 0.972i)7-s + (0.945 − 0.719i)8-s + (0.176 + 2.99i)9-s + (−3.09 − 0.167i)10-s + (−3.21 − 3.78i)11-s + (−2.21 − 0.741i)12-s + (−1.84 − 5.48i)13-s + (1.45 − 2.14i)14-s + (−1.51 − 2.50i)15-s + (−3.15 + 3.71i)16-s + (5.14 − 5.42i)17-s + ⋯ |
| L(s) = 1 | + (−1.26 + 0.278i)2-s + (−0.727 − 0.685i)3-s + (0.613 − 0.283i)4-s + (0.728 + 0.202i)5-s + (1.11 + 0.664i)6-s + (−0.387 + 0.367i)7-s + (0.334 − 0.254i)8-s + (0.0589 + 0.998i)9-s + (−0.977 − 0.0530i)10-s + (−0.970 − 1.14i)11-s + (−0.640 − 0.214i)12-s + (−0.512 − 1.52i)13-s + (0.388 − 0.572i)14-s + (−0.391 − 0.647i)15-s + (−0.789 + 0.929i)16-s + (1.24 − 1.31i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 177 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.189 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.231000 - 0.279894i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.231000 - 0.279894i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.26 + 1.18i)T \) |
| 59 | \( 1 + (-7.46 - 1.80i)T \) |
| good | 2 | \( 1 + (1.78 - 0.393i)T + (1.81 - 0.839i)T^{2} \) |
| 5 | \( 1 + (-1.62 - 0.452i)T + (4.28 + 2.57i)T^{2} \) |
| 7 | \( 1 + (1.02 - 0.972i)T + (0.378 - 6.98i)T^{2} \) |
| 11 | \( 1 + (3.21 + 3.78i)T + (-1.77 + 10.8i)T^{2} \) |
| 13 | \( 1 + (1.84 + 5.48i)T + (-10.3 + 7.86i)T^{2} \) |
| 17 | \( 1 + (-5.14 + 5.42i)T + (-0.920 - 16.9i)T^{2} \) |
| 19 | \( 1 + (0.166 + 0.418i)T + (-13.7 + 13.0i)T^{2} \) |
| 23 | \( 1 + (5.89 - 0.640i)T + (22.4 - 4.94i)T^{2} \) |
| 29 | \( 1 + (-0.131 + 0.597i)T + (-26.3 - 12.1i)T^{2} \) |
| 31 | \( 1 + (1.33 + 0.530i)T + (22.5 + 21.3i)T^{2} \) |
| 37 | \( 1 + (-6.08 + 7.99i)T + (-9.89 - 35.6i)T^{2} \) |
| 41 | \( 1 + (0.316 - 2.91i)T + (-40.0 - 8.81i)T^{2} \) |
| 43 | \( 1 + (-2.57 - 2.18i)T + (6.95 + 42.4i)T^{2} \) |
| 47 | \( 1 + (-0.731 - 2.63i)T + (-40.2 + 24.2i)T^{2} \) |
| 53 | \( 1 + (4.69 - 0.254i)T + (52.6 - 5.73i)T^{2} \) |
| 61 | \( 1 + (0.311 + 1.41i)T + (-55.3 + 25.6i)T^{2} \) |
| 67 | \( 1 + (3.77 + 4.97i)T + (-17.9 + 64.5i)T^{2} \) |
| 71 | \( 1 + (-5.93 + 1.64i)T + (60.8 - 36.6i)T^{2} \) |
| 73 | \( 1 + (-8.69 - 5.89i)T + (27.0 + 67.8i)T^{2} \) |
| 79 | \( 1 + (0.385 + 2.35i)T + (-74.8 + 25.2i)T^{2} \) |
| 83 | \( 1 + (3.14 + 5.92i)T + (-46.5 + 68.6i)T^{2} \) |
| 89 | \( 1 + (4.52 + 0.996i)T + (80.7 + 37.3i)T^{2} \) |
| 97 | \( 1 + (8.75 - 5.93i)T + (35.9 - 90.1i)T^{2} \) |
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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
−12.43652446369853238514115408847, −11.11829279775902474300658352809, −10.23321847047679905297906072870, −9.561354346931766912184146815011, −8.016514019289436155131576955244, −7.60973650315954625575008811586, −6.09343399968963091573824641998, −5.42762999750568253033290501345, −2.64789873024329929526492263106, −0.53894907768706987957990256881,
1.85077380490137143035245482223, 4.24478814681718138959158324053, 5.48766178788393584055903696724, 6.83651710297914185501814318976, 8.088672375678741458340544224465, 9.493588322080216491086336537250, 9.918125459904166596842024682489, 10.46999751919476091729338168514, 11.72753018323537375686732215939, 12.64634615680517620400240708643
