L(s) = 1 | + (−0.5 − 0.866i)2-s + (−1.61 + 2.80i)3-s + (−0.499 + 0.866i)4-s + (−0.572 − 0.991i)5-s + 3.23·6-s + (0.469 − 2.60i)7-s + 0.999·8-s + (−3.73 − 6.46i)9-s + (−0.572 + 0.991i)10-s + (0.425 − 0.736i)11-s + (−1.61 − 2.80i)12-s + 3.66·13-s + (−2.48 + 0.895i)14-s + 3.70·15-s + (−0.5 − 0.866i)16-s + (−0.0971 + 0.168i)17-s + ⋯ |
L(s) = 1 | + (−0.353 − 0.612i)2-s + (−0.933 + 1.61i)3-s + (−0.249 + 0.433i)4-s + (−0.255 − 0.443i)5-s + 1.32·6-s + (0.177 − 0.984i)7-s + 0.353·8-s + (−1.24 − 2.15i)9-s + (−0.180 + 0.313i)10-s + (0.128 − 0.222i)11-s + (−0.466 − 0.808i)12-s + 1.01·13-s + (−0.665 + 0.239i)14-s + 0.955·15-s + (−0.125 − 0.216i)16-s + (−0.0235 + 0.0408i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 406 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.970 + 0.239i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.777734 - 0.0944357i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.777734 - 0.0944357i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-0.469 + 2.60i)T \) |
| 29 | \( 1 + T \) |
good | 3 | \( 1 + (1.61 - 2.80i)T + (-1.5 - 2.59i)T^{2} \) |
| 5 | \( 1 + (0.572 + 0.991i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-0.425 + 0.736i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 3.66T + 13T^{2} \) |
| 17 | \( 1 + (0.0971 - 0.168i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.49 - 6.05i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.73 - 2.99i)T + (-11.5 + 19.9i)T^{2} \) |
| 31 | \( 1 + (-1.98 + 3.44i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3.60 + 6.24i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 4.04T + 41T^{2} \) |
| 43 | \( 1 - 9.77T + 43T^{2} \) |
| 47 | \( 1 + (0.0269 + 0.0466i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.33 + 2.31i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-4.02 + 6.96i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-2.54 - 4.40i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-5.80 + 10.0i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 11.2T + 71T^{2} \) |
| 73 | \( 1 + (-3.65 + 6.33i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-6.67 - 11.5i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 9.04T + 83T^{2} \) |
| 89 | \( 1 + (7.11 + 12.3i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 18.7T + 97T^{2} \) |
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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
−11.01828025524186905272067392081, −10.45547820212573166495554127702, −9.658350447534555451731085342971, −8.854264027063859698605163552300, −7.75558261345216626846731518240, −6.17692850348891812900497723862, −5.15537939946194177167017749331, −4.03677650483768899987299587188, −3.60425973440226220241552597735, −0.851976642188361233399330373561,
1.13178229940640331611444409566, 2.65987415526625686169579561304, 4.99819449311258280821761452212, 5.85498246569107409336890758711, 6.70823315345895789782052651889, 7.29524943579801342615447358423, 8.315276816009515401296647328322, 9.100794787017297194182954510050, 10.74187247024516940673574667487, 11.35728899210989105627566526535