L(s) = 1 | + 2.73i·2-s + (−0.813 + 1.52i)3-s − 5.49·4-s + (−4.18 − 2.22i)6-s + 0.851·7-s − 9.55i·8-s + (−1.67 − 2.48i)9-s − 3.31i·11-s + (4.46 − 8.39i)12-s − 3.60·13-s + 2.33i·14-s + 15.1·16-s + (6.81 − 4.58i)18-s − 8.71·19-s + (−0.692 + 1.30i)21-s + 9.07·22-s + ⋯ |
L(s) = 1 | + 1.93i·2-s + (−0.469 + 0.882i)3-s − 2.74·4-s + (−1.70 − 0.909i)6-s + 0.321·7-s − 3.37i·8-s + (−0.558 − 0.829i)9-s − 1.00i·11-s + (1.29 − 2.42i)12-s − 1.00·13-s + 0.623i·14-s + 3.79·16-s + (1.60 − 1.08i)18-s − 1.99·19-s + (−0.151 + 0.284i)21-s + 1.93·22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 429 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.115627 - 0.0288498i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.115627 - 0.0288498i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.813 - 1.52i)T \) |
| 11 | \( 1 + 3.31iT \) |
| 13 | \( 1 + 3.60T \) |
good | 2 | \( 1 - 2.73iT - 2T^{2} \) |
| 5 | \( 1 + 5T^{2} \) |
| 7 | \( 1 - 0.851T + 7T^{2} \) |
| 17 | \( 1 + 17T^{2} \) |
| 19 | \( 1 + 8.71T + 19T^{2} \) |
| 23 | \( 1 - 8.87iT - 23T^{2} \) |
| 29 | \( 1 + 29T^{2} \) |
| 31 | \( 1 - 31T^{2} \) |
| 37 | \( 1 - 37T^{2} \) |
| 41 | \( 1 + 7.49iT - 41T^{2} \) |
| 43 | \( 1 - 43T^{2} \) |
| 47 | \( 1 + 47T^{2} \) |
| 53 | \( 1 - 10.5iT - 53T^{2} \) |
| 59 | \( 1 + 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 67T^{2} \) |
| 71 | \( 1 + 71T^{2} \) |
| 73 | \( 1 + 4.44T + 73T^{2} \) |
| 79 | \( 1 - 79T^{2} \) |
| 83 | \( 1 + 0.671iT - 83T^{2} \) |
| 89 | \( 1 + 89T^{2} \) |
| 97 | \( 1 - 97T^{2} \) |
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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
−12.06105225629585745349174905776, −10.85672931108973935679231083522, −9.837723866989449823424273825398, −9.057586219152589030164742737816, −8.272363954659778385160317331893, −7.31367887876015349940804640199, −6.17539093940444220317573479247, −5.57681924593375342172460806433, −4.62649807274024841160591358179, −3.71778113766870582032869584778,
0.07877063027308025270644558124, 1.86521175740659871520923468378, 2.51476651640923192001494385426, 4.31878598861678942537632500469, 4.96533257109228075423794304769, 6.47013844736210560967151640720, 7.893356544307825903646893920188, 8.663159620456003292400367299604, 9.882262126481668275460664051118, 10.51037879226087502351616876076