L(s) = 1 | + (1.74 + 1.74i)2-s + (0.866 + 0.5i)3-s + 4.08i·4-s + (0.130 + 0.488i)5-s + (0.638 + 2.38i)6-s + (1.09 − 2.40i)7-s + (−3.63 + 3.63i)8-s + (0.499 + 0.866i)9-s + (−0.623 + 1.08i)10-s + (−1.54 − 5.75i)11-s + (−2.04 + 3.53i)12-s + (−3.50 + 0.833i)13-s + (6.11 − 2.29i)14-s + (−0.130 + 0.488i)15-s − 4.51·16-s + 0.933·17-s + ⋯ |
L(s) = 1 | + (1.23 + 1.23i)2-s + (0.499 + 0.288i)3-s + 2.04i·4-s + (0.0585 + 0.218i)5-s + (0.260 + 0.972i)6-s + (0.413 − 0.910i)7-s + (−1.28 + 1.28i)8-s + (0.166 + 0.288i)9-s + (−0.197 + 0.341i)10-s + (−0.464 − 1.73i)11-s + (−0.589 + 1.02i)12-s + (−0.972 + 0.231i)13-s + (1.63 − 0.613i)14-s + (−0.0337 + 0.126i)15-s − 1.12·16-s + 0.226·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 273 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.240 - 0.970i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.61472 + 2.06312i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.61472 + 2.06312i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (-1.09 + 2.40i)T \) |
| 13 | \( 1 + (3.50 - 0.833i)T \) |
good | 2 | \( 1 + (-1.74 - 1.74i)T + 2iT^{2} \) |
| 5 | \( 1 + (-0.130 - 0.488i)T + (-4.33 + 2.5i)T^{2} \) |
| 11 | \( 1 + (1.54 + 5.75i)T + (-9.52 + 5.5i)T^{2} \) |
| 17 | \( 1 - 0.933T + 17T^{2} \) |
| 19 | \( 1 + (7.66 + 2.05i)T + (16.4 + 9.5i)T^{2} \) |
| 23 | \( 1 - 8.12iT - 23T^{2} \) |
| 29 | \( 1 + (-1.96 - 3.40i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2.37 + 0.636i)T + (26.8 + 15.5i)T^{2} \) |
| 37 | \( 1 + (-0.859 + 0.859i)T - 37iT^{2} \) |
| 41 | \( 1 + (-7.84 - 2.10i)T + (35.5 + 20.5i)T^{2} \) |
| 43 | \( 1 + (0.152 + 0.0881i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-1.65 + 0.444i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-0.750 - 1.30i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (3.03 + 3.03i)T + 59iT^{2} \) |
| 61 | \( 1 + (6.74 - 3.89i)T + (30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (7.37 - 1.97i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + (-6.54 + 1.75i)T + (61.4 - 35.5i)T^{2} \) |
| 73 | \( 1 + (3.27 - 12.2i)T + (-63.2 - 36.5i)T^{2} \) |
| 79 | \( 1 + (-4.64 + 8.03i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-1.66 + 1.66i)T - 83iT^{2} \) |
| 89 | \( 1 + (-3.02 - 3.02i)T + 89iT^{2} \) |
| 97 | \( 1 + (0.856 + 3.19i)T + (-84.0 + 48.5i)T^{2} \) |
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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
−12.68457875043428256798394546252, −11.29295506393415527239118023856, −10.45536601544084226938690893870, −8.896904732847407968689786498112, −7.956822076413556367354376270147, −7.20529542245460047665612217243, −6.11799466443866347426318574214, −5.01683552888087636399535798911, −4.04947795358044632093590619363, −2.96765826466375887194223262718,
2.03411889615992003078403330498, 2.59523392982352595993081923159, 4.37234137437714351853570846022, 4.96861063931510702215245314954, 6.31253997271405055370129686431, 7.75559680255824639332437177643, 9.012408988518330521939101365284, 10.06958849587780270334993204554, 10.79888534714445479586509090745, 12.27329083285209411522204308282