L(s) = 1 | + (−1.35 + 0.783i)2-s + (0.226 − 0.391i)4-s + (−1.90 − 1.16i)5-s + (0.331 + 0.191i)7-s − 2.42i·8-s + (3.50 + 0.0880i)10-s + (2.30 + 3.99i)11-s + (3.53 − 0.731i)13-s − 0.598·14-s + (2.35 + 4.07i)16-s + (−4.29 − 2.47i)17-s + (−2.05 + 3.55i)19-s + (−0.888 + 0.483i)20-s + (−6.26 − 3.61i)22-s + (−5.72 + 3.30i)23-s + ⋯ |
L(s) = 1 | + (−0.959 + 0.553i)2-s + (0.113 − 0.195i)4-s + (−0.853 − 0.521i)5-s + (0.125 + 0.0722i)7-s − 0.856i·8-s + (1.10 + 0.0278i)10-s + (0.695 + 1.20i)11-s + (0.979 − 0.202i)13-s − 0.159·14-s + (0.587 + 1.01i)16-s + (−1.04 − 0.600i)17-s + (−0.470 + 0.815i)19-s + (−0.198 + 0.108i)20-s + (−1.33 − 0.770i)22-s + (−1.19 + 0.689i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.997 - 0.0762i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 585 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.997 - 0.0762i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0104628 + 0.274183i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0104628 + 0.274183i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 5 | \( 1 + (1.90 + 1.16i)T \) |
| 13 | \( 1 + (-3.53 + 0.731i)T \) |
good | 2 | \( 1 + (1.35 - 0.783i)T + (1 - 1.73i)T^{2} \) |
| 7 | \( 1 + (-0.331 - 0.191i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-2.30 - 3.99i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (4.29 + 2.47i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.05 - 3.55i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (5.72 - 3.30i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (2.65 + 4.59i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + 9.80T + 31T^{2} \) |
| 37 | \( 1 + (-1.44 + 0.835i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-6.22 - 10.7i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (7.10 + 4.10i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 2.71iT - 47T^{2} \) |
| 53 | \( 1 - 2.07iT - 53T^{2} \) |
| 59 | \( 1 + (4.11 - 7.12i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-1.07 + 1.86i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (10.1 - 5.87i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (-1.56 + 2.70i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 13.5iT - 73T^{2} \) |
| 79 | \( 1 + 1.45T + 79T^{2} \) |
| 83 | \( 1 - 2.02iT - 83T^{2} \) |
| 89 | \( 1 + (2.60 + 4.51i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (15.7 + 9.10i)T + (48.5 + 84.0i)T^{2} \) |
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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
−11.10391658739647751390345449462, −9.885683438132363713964786469538, −9.226180877731073910172570725097, −8.442365824875431466767767655948, −7.72071598274437113634292536425, −6.99140667909027683730616939640, −5.92423464760068019022736897743, −4.37619376889478485873184364657, −3.77104311375664227627251046401, −1.59706537978504086803786065596,
0.21874053838106304712147533808, 1.85746861710128448514313015621, 3.34655420387824710745313365974, 4.31762892583537443134196725281, 5.87357784108783039683802457252, 6.76596421423624238785413829111, 7.981841790776221242413072748718, 8.700908659668441233720015949517, 9.179509347828392650364978221491, 10.65005656573804870592331756898