L(s) = 1 | + 2.59i·2-s + (1.34 − 1.09i)3-s − 4.72·4-s + (0.626 + 1.08i)5-s + (2.84 + 3.47i)6-s + (−1.89 − 1.84i)7-s − 7.07i·8-s + (0.599 − 2.93i)9-s + (−2.81 + 1.62i)10-s + (−0.534 − 0.308i)11-s + (−6.34 + 5.17i)12-s + (1.06 + 0.613i)13-s + (4.78 − 4.92i)14-s + (2.02 + 0.769i)15-s + 8.88·16-s + (2.21 + 3.83i)17-s + ⋯ |
L(s) = 1 | + 1.83i·2-s + (0.774 − 0.632i)3-s − 2.36·4-s + (0.280 + 0.485i)5-s + (1.16 + 1.42i)6-s + (−0.717 − 0.696i)7-s − 2.50i·8-s + (0.199 − 0.979i)9-s + (−0.889 + 0.513i)10-s + (−0.161 − 0.0929i)11-s + (−1.83 + 1.49i)12-s + (0.294 + 0.170i)13-s + (1.27 − 1.31i)14-s + (0.523 + 0.198i)15-s + 2.22·16-s + (0.537 + 0.930i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 63 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.122 - 0.992i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.631977 + 0.714734i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.631977 + 0.714734i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (-1.34 + 1.09i)T \) |
| 7 | \( 1 + (1.89 + 1.84i)T \) |
good | 2 | \( 1 - 2.59iT - 2T^{2} \) |
| 5 | \( 1 + (-0.626 - 1.08i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.534 + 0.308i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.06 - 0.613i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (-2.21 - 3.83i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (1.64 + 0.950i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.11 - 2.37i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (5.07 - 2.93i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 - 2.48iT - 31T^{2} \) |
| 37 | \( 1 + (-1.33 + 2.30i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (2.09 - 3.63i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (2.24 + 3.89i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 - 7.61T + 47T^{2} \) |
| 53 | \( 1 + (2.67 - 1.54i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 3.56T + 59T^{2} \) |
| 61 | \( 1 - 14.4iT - 61T^{2} \) |
| 67 | \( 1 - 13.6T + 67T^{2} \) |
| 71 | \( 1 + 10.4iT - 71T^{2} \) |
| 73 | \( 1 + (-9.95 + 5.74i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 4.03T + 79T^{2} \) |
| 83 | \( 1 + (4.36 + 7.56i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-0.811 + 1.40i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.76 + 5.06i)T + (48.5 - 84.0i)T^{2} \) |
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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
−15.16588689974219946434001755978, −14.29231308923270525622676630516, −13.58366561354356241786845569736, −12.68613160375460222643435963050, −10.20610073482877297815834451708, −8.993459269673165552108802506432, −7.85983545816181433438674536387, −6.88988462070366318125480203112, −6.00749317218271082949108148903, −3.80392450691408923662945055485,
2.38689141958865116276646550313, 3.72548420776915063705268294546, 5.24996934162755556475000929905, 8.339534536834143685160529088486, 9.406325629227750639468486772485, 9.938354692208885847072767682546, 11.23722918398700089411391213770, 12.49784800175515714895548963143, 13.24825193753315340477253426123, 14.26118856975441911087095115445