| L(s) = 1 | + (0.452 − 1.33i)2-s + (−2.61 + 0.196i)3-s + (−1.58 − 1.21i)4-s + (1.39 + 2.03i)5-s + (−0.922 + 3.59i)6-s + (−2.07 − 1.64i)7-s + (−2.34 + 1.58i)8-s + (3.84 − 0.579i)9-s + (3.36 − 0.939i)10-s + (−0.163 + 1.08i)11-s + (4.39 + 2.86i)12-s + (5.03 + 4.01i)13-s + (−3.13 + 2.03i)14-s + (−4.03 − 5.06i)15-s + (1.05 + 3.85i)16-s + (0.946 − 0.292i)17-s + ⋯ |
| L(s) = 1 | + (0.320 − 0.947i)2-s + (−1.51 + 0.113i)3-s + (−0.794 − 0.606i)4-s + (0.621 + 0.911i)5-s + (−0.376 + 1.46i)6-s + (−0.784 − 0.620i)7-s + (−0.829 + 0.558i)8-s + (1.28 − 0.193i)9-s + (1.06 − 0.297i)10-s + (−0.0492 + 0.326i)11-s + (1.26 + 0.826i)12-s + (1.39 + 1.11i)13-s + (−0.838 + 0.544i)14-s + (−1.04 − 1.30i)15-s + (0.263 + 0.964i)16-s + (0.229 − 0.0708i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 392 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0237i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.849828 + 0.0101072i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.849828 + 0.0101072i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.452 + 1.33i)T \) |
| 7 | \( 1 + (2.07 + 1.64i)T \) |
| good | 3 | \( 1 + (2.61 - 0.196i)T + (2.96 - 0.447i)T^{2} \) |
| 5 | \( 1 + (-1.39 - 2.03i)T + (-1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.163 - 1.08i)T + (-10.5 - 3.24i)T^{2} \) |
| 13 | \( 1 + (-5.03 - 4.01i)T + (2.89 + 12.6i)T^{2} \) |
| 17 | \( 1 + (-0.946 + 0.292i)T + (14.0 - 9.57i)T^{2} \) |
| 19 | \( 1 + (-1.98 - 1.14i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-7.19 - 2.21i)T + (19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (9.74 + 2.22i)T + (26.1 + 12.5i)T^{2} \) |
| 31 | \( 1 + (-2.52 - 4.37i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.75 - 2.97i)T + (-2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (1.95 + 0.942i)T + (25.5 + 32.0i)T^{2} \) |
| 43 | \( 1 + (-2.25 - 4.68i)T + (-26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-1.94 - 4.95i)T + (-34.4 + 31.9i)T^{2} \) |
| 53 | \( 1 + (-3.29 - 3.55i)T + (-3.96 + 52.8i)T^{2} \) |
| 59 | \( 1 + (4.36 - 6.39i)T + (-21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-8.57 + 9.24i)T + (-4.55 - 60.8i)T^{2} \) |
| 67 | \( 1 + (10.4 - 6.00i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (0.833 + 3.65i)T + (-63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (4.24 - 10.8i)T + (-53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (-2.96 + 5.13i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (1.96 - 1.57i)T + (18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-3.19 + 0.481i)T + (85.0 - 26.2i)T^{2} \) |
| 97 | \( 1 - 8.96T + 97T^{2} \) |
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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
−11.15356184320055794908759042894, −10.70147667106789702972121824601, −9.919399560031759582803010970546, −9.114068644956447325283092249090, −7.06269296916848728237592773434, −6.30305777229812577819501780870, −5.56718113055080056039656908501, −4.30519991767961797795670579777, −3.18780210750811610665861097572, −1.33628073357104645517861548237,
0.71587332304501023339504527379, 3.47064848974148904631841914802, 5.09075103964669403596404270637, 5.62150570266428318918667393318, 6.13170262581377067731157373414, 7.21052015081554520173771281483, 8.620970108746508721793661468019, 9.240528115777347623976213710121, 10.43668850839883963106057858218, 11.46404554077352861069769398029
