L(s) = 1 | + (1.36 + 2.36i)2-s + (−2.73 + 4.73i)4-s + (−0.5 + 0.866i)5-s + (0.633 + 1.09i)7-s − 9.46·8-s − 2.73·10-s + (1.13 + 1.96i)11-s + (2.73 − 4.73i)13-s + (−1.73 + 3i)14-s + (−7.46 − 12.9i)16-s + 0.732·17-s − 2.46·19-s + (−2.73 − 4.73i)20-s + (−3.09 + 5.36i)22-s + (−1.73 + 3i)23-s + ⋯ |
L(s) = 1 | + (0.965 + 1.67i)2-s + (−1.36 + 2.36i)4-s + (−0.223 + 0.387i)5-s + (0.239 + 0.415i)7-s − 3.34·8-s − 0.863·10-s + (0.341 + 0.592i)11-s + (0.757 − 1.31i)13-s + (−0.462 + 0.801i)14-s + (−1.86 − 3.23i)16-s + 0.177·17-s − 0.565·19-s + (−0.610 − 1.05i)20-s + (−0.660 + 1.14i)22-s + (−0.361 + 0.625i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 405 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.984 + 0.173i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.165825 - 1.89539i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.165825 - 1.89539i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
good | 2 | \( 1 + (-1.36 - 2.36i)T + (-1 + 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.633 - 1.09i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-1.13 - 1.96i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-2.73 + 4.73i)T + (-6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.732T + 17T^{2} \) |
| 19 | \( 1 + 2.46T + 19T^{2} \) |
| 23 | \( 1 + (1.73 - 3i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (-3.59 - 6.23i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-1.5 + 2.59i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 0.732T + 37T^{2} \) |
| 41 | \( 1 + (1.59 - 2.76i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.09 - 8.83i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-2.63 - 4.56i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 - 3.26T + 53T^{2} \) |
| 59 | \( 1 + (-5.86 + 10.1i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (2 + 3.46i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.73 + 3i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 0.267T + 71T^{2} \) |
| 73 | \( 1 - 9.66T + 73T^{2} \) |
| 79 | \( 1 + (4.26 + 7.39i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (-4.09 - 7.09i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + 5.19T + 89T^{2} \) |
| 97 | \( 1 + (3.83 + 6.63i)T + (-48.5 + 84.0i)T^{2} \) |
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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
−12.18207642515705751637940758412, −11.02205020827738593807648067126, −9.603806174748282933469960518479, −8.428848140622265578115970261404, −7.899804467210816922651734806742, −6.88765769128456567961206306058, −6.04479816052292695329768466713, −5.19779654704415700788058986464, −4.10165376231819603599411643757, −3.04367738719417493586361704995,
1.01053617757719889937601477821, 2.35090648063785735326048693140, 3.90719652756023183783281195154, 4.28105763848237623438696308070, 5.57693200434407207387072893433, 6.59904456475014027428731226369, 8.488341614347387466734063390495, 9.180599278617522443174533777525, 10.30567940076423927479461632115, 10.94902446248956599800103503468