L(s) = 1 | + (0.669 − 1.24i)2-s + (1.94 − 1.12i)3-s + (−1.10 − 1.66i)4-s + (−0.5 + 0.866i)5-s + (−0.0964 − 3.17i)6-s + (2.47 − 0.947i)7-s + (−2.81 + 0.257i)8-s + (1.02 − 1.78i)9-s + (0.743 + 1.20i)10-s + (0.656 + 1.13i)11-s + (−4.02 − 2.00i)12-s − 3.02·13-s + (0.473 − 3.71i)14-s + 2.24i·15-s + (−1.56 + 3.68i)16-s + (−0.313 + 0.181i)17-s + ⋯ |
L(s) = 1 | + (0.473 − 0.880i)2-s + (1.12 − 0.649i)3-s + (−0.551 − 0.834i)4-s + (−0.223 + 0.387i)5-s + (−0.0393 − 1.29i)6-s + (0.933 − 0.358i)7-s + (−0.995 + 0.0908i)8-s + (0.342 − 0.593i)9-s + (0.235 + 0.380i)10-s + (0.198 + 0.342i)11-s + (−1.16 − 0.579i)12-s − 0.838·13-s + (0.126 − 0.991i)14-s + 0.580i·15-s + (−0.391 + 0.920i)16-s + (−0.0760 + 0.0439i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.191 + 0.981i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.33702 - 1.62345i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.33702 - 1.62345i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.669 + 1.24i)T \) |
| 5 | \( 1 + (0.5 - 0.866i)T \) |
| 7 | \( 1 + (-2.47 + 0.947i)T \) |
good | 3 | \( 1 + (-1.94 + 1.12i)T + (1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (-0.656 - 1.13i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + 3.02T + 13T^{2} \) |
| 17 | \( 1 + (0.313 - 0.181i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.16 - 1.82i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (5.74 + 3.31i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 3.35iT - 29T^{2} \) |
| 31 | \( 1 + (-3.37 - 5.84i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.89 - 2.24i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 4.07iT - 41T^{2} \) |
| 43 | \( 1 + 5.01T + 43T^{2} \) |
| 47 | \( 1 + (-6.23 + 10.7i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (2.39 - 1.38i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (11.5 - 6.65i)T + (29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-5.04 + 8.74i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (0.897 + 1.55i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 12.4iT - 71T^{2} \) |
| 73 | \( 1 + (-7.83 + 4.52i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (7.89 + 4.55i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 10.7iT - 83T^{2} \) |
| 89 | \( 1 + (1.99 + 1.15i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 2.74iT - 97T^{2} \) |
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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
−11.85176533174357036326136818433, −10.68857291598885285988358274423, −9.866214839796621001386660790149, −8.680306823806479268620118127542, −7.84957345988921292968611521081, −6.83327232335454368099034715462, −5.17070942302301418808997587206, −3.98532069270125675567007411235, −2.74114935916115363312703918790, −1.66498923035021734309172565933,
2.66849595766888831062520319716, 4.00386942445977429398438048123, 4.81921313242587118814882747453, 5.99622675462267818793756241440, 7.69489340226522404672886839308, 8.063257971561765723764662168832, 9.131448041764528916233564847409, 9.709537233807456712662150082934, 11.46654039600738424453955420646, 12.17237733690818695938769530748