L(s) = 1 | + (−1.05 + 0.609i)2-s + (−0.256 + 0.443i)4-s + i·5-s + (3.11 + 1.80i)7-s − 3.06i·8-s + (−0.609 − 1.05i)10-s + (4.65 − 2.68i)11-s + (1.81 − 3.11i)13-s − 4.39·14-s + (1.35 + 2.34i)16-s + (0.565 − 0.980i)17-s + (−1.96 − 1.13i)19-s + (−0.443 − 0.256i)20-s + (−3.27 + 5.67i)22-s + (1.94 + 3.37i)23-s + ⋯ |
L(s) = 1 | + (−0.746 + 0.431i)2-s + (−0.128 + 0.221i)4-s + 0.447i·5-s + (1.17 + 0.680i)7-s − 1.08i·8-s + (−0.192 − 0.334i)10-s + (1.40 − 0.809i)11-s + (0.504 − 0.863i)13-s − 1.17·14-s + (0.339 + 0.587i)16-s + (0.137 − 0.237i)17-s + (−0.450 − 0.260i)19-s + (−0.0992 − 0.0572i)20-s + (−0.698 + 1.20i)22-s + (0.405 + 0.702i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.515 - 0.856i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.00042 + 0.565447i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.00042 + 0.565447i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 - iT \) |
| 13 | \( 1 + (-1.81 + 3.11i)T \) |
good | 2 | \( 1 + (1.05 - 0.609i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-3.11 - 1.80i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-4.65 + 2.68i)T + (5.5 - 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.565 + 0.980i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (1.96 + 1.13i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.94 - 3.37i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (0.0123 + 0.0214i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 - 5.46iT - 31T^{2} \) |
| 37 | \( 1 + (-7.53 + 4.35i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.23 - 1.86i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (0.565 - 0.980i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 - 2.58iT - 47T^{2} \) |
| 53 | \( 1 - 4.43T + 53T^{2} \) |
| 59 | \( 1 + (-0.148 - 0.0857i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (1.68 - 2.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.54 + 3.19i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (9.35 + 5.39i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 4.70iT - 73T^{2} \) |
| 79 | \( 1 + 11.9T + 79T^{2} \) |
| 83 | \( 1 - 12.1iT - 83T^{2} \) |
| 89 | \( 1 + (13.9 - 8.07i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (10.5 + 6.08i)T + (48.5 + 84.0i)T^{2} \) |
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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
−10.91460215919695323220156172602, −9.720346077584528605635621661742, −8.816240910419103443217280835452, −8.377989568207921333825491993155, −7.46854934902979942473585719179, −6.49310403351457170257459829511, −5.52829986860158000082065188189, −4.16340033708136951503235893315, −3.07043982705124018575311043362, −1.23550543909298884765894061055,
1.15885647089443219427386733001, 1.93934773045573340028326406107, 4.16208190302162688224111251859, 4.66027143459471635066458444635, 6.03021035033725017364334153258, 7.15066189096798653861878306373, 8.246390820032673773223738726590, 8.851911542890569506097048168131, 9.679195642188487370880048480268, 10.47037313948120179358462678912