L(s) = 1 | + (0.809 − 0.587i)2-s + (−1.67 + 0.445i)3-s + (0.309 − 0.951i)4-s + (1.77 − 2.44i)5-s + (−1.09 + 1.34i)6-s + (−0.951 − 0.309i)7-s + (−0.309 − 0.951i)8-s + (2.60 − 1.49i)9-s − 3.02i·10-s + (−2.94 − 1.51i)11-s + (−0.0930 + 1.72i)12-s + (1.46 + 2.01i)13-s + (−0.951 + 0.309i)14-s + (−1.88 + 4.88i)15-s + (−0.809 − 0.587i)16-s + (−2.63 − 1.91i)17-s + ⋯ |
L(s) = 1 | + (0.572 − 0.415i)2-s + (−0.966 + 0.257i)3-s + (0.154 − 0.475i)4-s + (0.794 − 1.09i)5-s + (−0.445 + 0.548i)6-s + (−0.359 − 0.116i)7-s + (−0.109 − 0.336i)8-s + (0.867 − 0.497i)9-s − 0.955i·10-s + (−0.889 − 0.457i)11-s + (−0.0268 + 0.499i)12-s + (0.406 + 0.559i)13-s + (−0.254 + 0.0825i)14-s + (−0.486 + 1.26i)15-s + (−0.202 − 0.146i)16-s + (−0.640 − 0.465i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 462 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 + 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.706968 - 1.13892i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.706968 - 1.13892i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.809 + 0.587i)T \) |
| 3 | \( 1 + (1.67 - 0.445i)T \) |
| 7 | \( 1 + (0.951 + 0.309i)T \) |
| 11 | \( 1 + (2.94 + 1.51i)T \) |
good | 5 | \( 1 + (-1.77 + 2.44i)T + (-1.54 - 4.75i)T^{2} \) |
| 13 | \( 1 + (-1.46 - 2.01i)T + (-4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (2.63 + 1.91i)T + (5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (2.06 - 0.671i)T + (15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 7.69iT - 23T^{2} \) |
| 29 | \( 1 + (-2.83 + 8.71i)T + (-23.4 - 17.0i)T^{2} \) |
| 31 | \( 1 + (-5.14 + 3.73i)T + (9.57 - 29.4i)T^{2} \) |
| 37 | \( 1 + (-1.36 + 4.21i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-3.00 - 9.25i)T + (-33.1 + 24.0i)T^{2} \) |
| 43 | \( 1 - 7.49iT - 43T^{2} \) |
| 47 | \( 1 + (9.50 - 3.08i)T + (38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (-5.92 - 8.15i)T + (-16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-4.10 - 1.33i)T + (47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (1.82 - 2.51i)T + (-18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 11.5T + 67T^{2} \) |
| 71 | \( 1 + (-6.23 + 8.57i)T + (-21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (5.37 + 1.74i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (1.33 + 1.83i)T + (-24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-7.58 - 5.51i)T + (25.6 + 78.9i)T^{2} \) |
| 89 | \( 1 - 11.7iT - 89T^{2} \) |
| 97 | \( 1 + (-10.8 + 7.87i)T + (29.9 - 92.2i)T^{2} \) |
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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
−10.86992561151836626854117499330, −9.986585536782417513624971453584, −9.314364546854584847658672455204, −8.150519758053704853433683132069, −6.42544007752752608563234594321, −6.02059388803830180108482019103, −4.83670141164226732339237386071, −4.30990708770602954055927274140, −2.44838153579977043109053718087, −0.77140389369839957222872762188,
2.12252282115464081919050881059, 3.43876959095025844886566097629, 5.02258520768342761282139121493, 5.73884380821749932232936731659, 6.67001806485769688447911129361, 7.14346626332954285539837010364, 8.424034745758781007673361605804, 9.960135803690426365603624956590, 10.52024534954088695405366352324, 11.26720671497436204673412189364