L(s) = 1 | + (0.642 − 0.766i)2-s + (0.173 − 0.984i)3-s + (−0.173 − 0.984i)4-s + (−0.642 − 0.766i)6-s + (−0.866 − 0.500i)8-s + (−0.939 − 0.342i)9-s − 12-s + (−1.10 − 0.296i)13-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.499i)18-s + (−0.866 + 0.5i)23-s + (−0.642 + 0.766i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 0.657i)26-s + (−0.5 + 0.866i)27-s + ⋯ |
L(s) = 1 | + (0.642 − 0.766i)2-s + (0.173 − 0.984i)3-s + (−0.173 − 0.984i)4-s + (−0.642 − 0.766i)6-s + (−0.866 − 0.500i)8-s + (−0.939 − 0.342i)9-s − 12-s + (−1.10 − 0.296i)13-s + (−0.939 + 0.342i)16-s + (−0.866 + 0.499i)18-s + (−0.866 + 0.5i)23-s + (−0.642 + 0.766i)24-s + (−0.866 − 0.5i)25-s + (−0.939 + 0.657i)26-s + (−0.5 + 0.866i)27-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 3312 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.843 - 0.537i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.201044540\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.201044540\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.642 + 0.766i)T \) |
| 3 | \( 1 + (-0.173 + 0.984i)T \) |
| 23 | \( 1 + (0.866 - 0.5i)T \) |
good | 5 | \( 1 + (0.866 + 0.5i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 11 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 13 | \( 1 + (1.10 + 0.296i)T + (0.866 + 0.5i)T^{2} \) |
| 17 | \( 1 - T^{2} \) |
| 19 | \( 1 + iT^{2} \) |
| 29 | \( 1 + (0.0451 + 0.168i)T + (-0.866 + 0.5i)T^{2} \) |
| 31 | \( 1 + (0.642 + 1.11i)T + (-0.5 + 0.866i)T^{2} \) |
| 37 | \( 1 - iT^{2} \) |
| 41 | \( 1 + (-1.62 + 0.939i)T + (0.5 - 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.866 + 0.5i)T^{2} \) |
| 47 | \( 1 + (-0.939 + 1.62i)T + (-0.5 - 0.866i)T^{2} \) |
| 53 | \( 1 - iT^{2} \) |
| 59 | \( 1 + (-0.5 + 1.86i)T + (-0.866 - 0.5i)T^{2} \) |
| 61 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 67 | \( 1 + (-0.866 - 0.5i)T^{2} \) |
| 71 | \( 1 + 1.28iT - T^{2} \) |
| 73 | \( 1 - 1.53iT - T^{2} \) |
| 79 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.866 - 0.5i)T^{2} \) |
| 89 | \( 1 + T^{2} \) |
| 97 | \( 1 + (0.5 + 0.866i)T^{2} \) |
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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
−8.289323430700708090402444899517, −7.56537794847869749166356936423, −6.87343198004343168787301148949, −5.83745694012777582669235384648, −5.53809646955414984395217723419, −4.33038565878221517071587559923, −3.55145603458386748570029644950, −2.42148530167617330499323008341, −2.00931072359312939254148980709, −0.51779728772103489763958206303,
2.30559670707499222327654737909, 3.11484178502334324256748151399, 4.12614550354495853281006534681, 4.54447724802833425419431304318, 5.48649353698896184449417528878, 5.98448577331832638682053041467, 7.07108441249072634804424649391, 7.68645379013592894576874928498, 8.462655695451532459853425323193, 9.207437767888112689484546169721