| L(s) = 1 | + (0.168 + 0.114i)2-s + (2.17 + 2.01i)3-s + (−0.715 − 1.82i)4-s + (0.954 − 0.294i)5-s + (0.134 + 0.588i)6-s + (−0.383 + 2.61i)7-s + (0.179 − 0.786i)8-s + (0.432 + 5.77i)9-s + (0.194 + 0.0599i)10-s + (−0.109 + 1.45i)11-s + (2.12 − 5.40i)12-s + (−0.900 + 0.433i)13-s + (−0.364 + 0.396i)14-s + (2.66 + 1.28i)15-s + (−2.75 + 2.55i)16-s + (4.78 − 0.721i)17-s + ⋯ |
| L(s) = 1 | + (0.119 + 0.0811i)2-s + (1.25 + 1.16i)3-s + (−0.357 − 0.911i)4-s + (0.427 − 0.131i)5-s + (0.0548 + 0.240i)6-s + (−0.145 + 0.989i)7-s + (0.0634 − 0.277i)8-s + (0.144 + 1.92i)9-s + (0.0615 + 0.0189i)10-s + (−0.0329 + 0.439i)11-s + (0.612 − 1.56i)12-s + (−0.249 + 0.120i)13-s + (−0.0975 + 0.105i)14-s + (0.689 + 0.331i)15-s + (−0.687 + 0.638i)16-s + (1.16 − 0.175i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.416 - 0.908i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 637 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.416 - 0.908i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.94439 + 1.24720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.94439 + 1.24720i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 7 | \( 1 + (0.383 - 2.61i)T \) |
| 13 | \( 1 + (0.900 - 0.433i)T \) |
| good | 2 | \( 1 + (-0.168 - 0.114i)T + (0.730 + 1.86i)T^{2} \) |
| 3 | \( 1 + (-2.17 - 2.01i)T + (0.224 + 2.99i)T^{2} \) |
| 5 | \( 1 + (-0.954 + 0.294i)T + (4.13 - 2.81i)T^{2} \) |
| 11 | \( 1 + (0.109 - 1.45i)T + (-10.8 - 1.63i)T^{2} \) |
| 17 | \( 1 + (-4.78 + 0.721i)T + (16.2 - 5.01i)T^{2} \) |
| 19 | \( 1 + (-0.372 - 0.645i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.24 - 0.639i)T + (21.9 + 6.77i)T^{2} \) |
| 29 | \( 1 + (-1.92 - 2.41i)T + (-6.45 + 28.2i)T^{2} \) |
| 31 | \( 1 + (-3.10 + 5.38i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (-2.27 + 5.78i)T + (-27.1 - 25.1i)T^{2} \) |
| 41 | \( 1 + (-1.39 + 6.11i)T + (-36.9 - 17.7i)T^{2} \) |
| 43 | \( 1 + (0.925 + 4.05i)T + (-38.7 + 18.6i)T^{2} \) |
| 47 | \( 1 + (1.59 + 1.08i)T + (17.1 + 43.7i)T^{2} \) |
| 53 | \( 1 + (1.35 + 3.44i)T + (-38.8 + 36.0i)T^{2} \) |
| 59 | \( 1 + (4.50 + 1.39i)T + (48.7 + 33.2i)T^{2} \) |
| 61 | \( 1 + (1.97 - 5.02i)T + (-44.7 - 41.4i)T^{2} \) |
| 67 | \( 1 + (-6.31 + 10.9i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (6.88 - 8.62i)T + (-15.7 - 69.2i)T^{2} \) |
| 73 | \( 1 + (12.5 - 8.58i)T + (26.6 - 67.9i)T^{2} \) |
| 79 | \( 1 + (1.38 + 2.40i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (13.0 + 6.29i)T + (51.7 + 64.8i)T^{2} \) |
| 89 | \( 1 + (0.106 + 1.41i)T + (-88.0 + 13.2i)T^{2} \) |
| 97 | \( 1 + 3.09T + 97T^{2} \) |
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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
−10.22007276652397425529083453539, −9.769079786024473474976926716240, −9.193069439317747698962171628408, −8.540571476804298477001558640870, −7.36297714514507852441320071132, −5.82928242442979208150842562240, −5.20002631447283716081886159149, −4.24506244577004722893540908715, −3.05164834892038636248622273776, −1.92874307922939694402390383442,
1.21225294652873303119678659587, 2.80492738628506726648627489200, 3.32231292441762437515525659537, 4.58246636485738020882358773287, 6.25889466225129070035826098597, 7.16299264555175886717989471067, 7.86343239628398125475884522777, 8.380141758036243285550646610819, 9.389739917832525532425903879045, 10.20506918589248318165254314842
