L(s) = 1 | + (1.95 + 1.41i)2-s + (−0.874 + 2.68i)3-s + (1.18 + 3.64i)4-s + (0.809 − 0.587i)5-s + (−5.52 + 4.01i)6-s + (0.618 + 1.90i)7-s + (−1.36 + 4.19i)8-s + (−4.04 − 2.93i)9-s + 2.41·10-s − 10.8·12-s + (−0.947 − 0.688i)13-s + (−1.49 + 4.59i)14-s + (0.874 + 2.68i)15-s + (−2.42 + 1.76i)16-s + (5.52 − 4.01i)17-s + (−3.73 − 11.4i)18-s + ⋯ |
L(s) = 1 | + (1.38 + 1.00i)2-s + (−0.504 + 1.55i)3-s + (0.591 + 1.82i)4-s + (0.361 − 0.262i)5-s + (−2.25 + 1.63i)6-s + (0.233 + 0.718i)7-s + (−0.482 + 1.48i)8-s + (−1.34 − 0.979i)9-s + 0.763·10-s − 3.12·12-s + (−0.262 − 0.190i)13-s + (−0.398 + 1.22i)14-s + (0.225 + 0.694i)15-s + (−0.606 + 0.440i)16-s + (1.33 − 0.973i)17-s + (−0.879 − 2.70i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.999 - 0.0439i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 605 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.999 - 0.0439i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.0610965 + 2.77847i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.0610965 + 2.77847i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-0.809 + 0.587i)T \) |
| 11 | \( 1 \) |
good | 2 | \( 1 + (-1.95 - 1.41i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.874 - 2.68i)T + (-2.42 - 1.76i)T^{2} \) |
| 7 | \( 1 + (-0.618 - 1.90i)T + (-5.66 + 4.11i)T^{2} \) |
| 13 | \( 1 + (0.947 + 0.688i)T + (4.01 + 12.3i)T^{2} \) |
| 17 | \( 1 + (-5.52 + 4.01i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-15.3 - 11.1i)T^{2} \) |
| 23 | \( 1 + 2.82T + 23T^{2} \) |
| 29 | \( 1 + (-1.13 - 3.47i)T + (-23.4 + 17.0i)T^{2} \) |
| 31 | \( 1 + (9.57 + 29.4i)T^{2} \) |
| 37 | \( 1 + (2.36 + 7.28i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.85 - 5.70i)T + (-33.1 - 24.0i)T^{2} \) |
| 43 | \( 1 - 6T + 43T^{2} \) |
| 47 | \( 1 + (-0.874 + 2.68i)T + (-38.0 - 27.6i)T^{2} \) |
| 53 | \( 1 + (9.43 + 6.85i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-0.511 - 1.57i)T + (-47.7 + 34.6i)T^{2} \) |
| 61 | \( 1 + (7.53 - 5.47i)T + (18.8 - 58.0i)T^{2} \) |
| 67 | \( 1 - 12.4T + 67T^{2} \) |
| 71 | \( 1 + (9.15 - 6.65i)T + (21.9 - 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.362 - 1.11i)T + (-59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (-3.23 - 2.35i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (4.85 - 3.52i)T + (25.6 - 78.9i)T^{2} \) |
| 89 | \( 1 + 13.3T + 89T^{2} \) |
| 97 | \( 1 + (2.95 + 2.14i)T + (29.9 + 92.2i)T^{2} \) |
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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
−11.27953419137247928616079895441, −10.13124797049374649105738159640, −9.437486846050808058620891821565, −8.386210427298731981370624195290, −7.26648583809323972309375506366, −5.98691769864380659013618842391, −5.40154445623970323009605813147, −4.90937195285153085844969973365, −3.91326797233694192384758085417, −2.90424192660060011062777369118,
1.19155571365831732829406612797, 2.06582526720419860497490719606, 3.30850047495763793201893755807, 4.54144613373311084393936136669, 5.74139466518471457190306096110, 6.22930347898547439214780829182, 7.30119671691267531254755648238, 8.116106854384740686694701638734, 9.886544928560333133178779953205, 10.65142318295026199332144910575