L(s) = 1 | + (1.31 + 0.518i)2-s + (1.46 + 1.36i)4-s + (2.03 + 1.17i)5-s + (2.03 − 1.69i)7-s + (1.21 + 2.55i)8-s + (2.06 + 2.59i)10-s + (−2.18 + 1.26i)11-s − 1.48i·13-s + (3.55 − 1.17i)14-s + (0.274 + 3.99i)16-s + (1.66 − 0.958i)17-s + (−0.454 + 0.786i)19-s + (1.36 + 4.48i)20-s + (−3.53 + 0.526i)22-s + (−4.55 − 2.63i)23-s + ⋯ |
L(s) = 1 | + (0.930 + 0.366i)2-s + (0.730 + 0.682i)4-s + (0.908 + 0.524i)5-s + (0.768 − 0.639i)7-s + (0.429 + 0.902i)8-s + (0.652 + 0.821i)10-s + (−0.659 + 0.380i)11-s − 0.411i·13-s + (0.949 − 0.312i)14-s + (0.0687 + 0.997i)16-s + (0.402 − 0.232i)17-s + (−0.104 + 0.180i)19-s + (0.306 + 1.00i)20-s + (−0.753 + 0.112i)22-s + (−0.950 − 0.548i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.649 - 0.759i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 756 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.649 - 0.759i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(2.93365 + 1.35122i\) |
\(L(\frac12)\) |
\(\approx\) |
\(2.93365 + 1.35122i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-1.31 - 0.518i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 + (-2.03 + 1.69i)T \) |
good | 5 | \( 1 + (-2.03 - 1.17i)T + (2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (2.18 - 1.26i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + 1.48iT - 13T^{2} \) |
| 17 | \( 1 + (-1.66 + 0.958i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (0.454 - 0.786i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (4.55 + 2.63i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 4.85T + 29T^{2} \) |
| 31 | \( 1 + (3.77 + 6.53i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (4.63 - 8.03i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 12.0iT - 41T^{2} \) |
| 43 | \( 1 + 10.9iT - 43T^{2} \) |
| 47 | \( 1 + (2.04 - 3.54i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.83 + 4.91i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-3.98 - 6.90i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.72 - 3.30i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (1.36 - 0.790i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 0.670iT - 71T^{2} \) |
| 73 | \( 1 + (-2.49 + 1.44i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (13.5 + 7.81i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 6.82T + 83T^{2} \) |
| 89 | \( 1 + (-3.23 - 1.86i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 14.0iT - 97T^{2} \) |
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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
−10.35970237074315533529777313361, −10.04267811981243405519549804431, −8.384325986610858285256612764109, −7.73698265951537758376126444331, −6.83092342334353479600706399930, −5.96207063825652413129806741832, −5.11275527173969075183449625933, −4.20596427319332675624224647380, −2.91056373188688390274417916548, −1.88615000324644046750093233227,
1.54089689763529319272849295872, 2.40495883439777950738762246981, 3.76271968267499229149568858800, 5.07644127046583940941520937534, 5.44534948886432367312116728336, 6.33880340141263588034984605722, 7.56115155555681757268284253495, 8.632036810694177343373974591788, 9.491379306351214328893029118188, 10.41212096419764907835631040944