L(s) = 1 | + (−1.11 + 1.93i)5-s + (−0.366 + 1.36i)7-s + (3.87 − 2.23i)11-s + (−1.09 − 4.09i)13-s + (2.23 − 2.23i)17-s − 2i·19-s + (3.05 − 0.818i)23-s + (−2.5 − 4.33i)25-s + (2.23 + 3.87i)29-s + (−2 + 3.46i)31-s + (−2.23 − 2.23i)35-s + (−3 − 3i)37-s + (7.74 + 4.47i)41-s + (4.09 + 1.09i)43-s + (9.16 + 2.45i)47-s + ⋯ |
L(s) = 1 | + (−0.499 + 0.866i)5-s + (−0.138 + 0.516i)7-s + (1.16 − 0.674i)11-s + (−0.304 − 1.13i)13-s + (0.542 − 0.542i)17-s − 0.458i·19-s + (0.636 − 0.170i)23-s + (−0.5 − 0.866i)25-s + (0.415 + 0.719i)29-s + (−0.359 + 0.622i)31-s + (−0.377 − 0.377i)35-s + (−0.493 − 0.493i)37-s + (1.20 + 0.698i)41-s + (0.624 + 0.167i)43-s + (1.33 + 0.358i)47-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1620 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.979 - 0.203i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.619709473\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.619709473\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (1.11 - 1.93i)T \) |
good | 7 | \( 1 + (0.366 - 1.36i)T + (-6.06 - 3.5i)T^{2} \) |
| 11 | \( 1 + (-3.87 + 2.23i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (1.09 + 4.09i)T + (-11.2 + 6.5i)T^{2} \) |
| 17 | \( 1 + (-2.23 + 2.23i)T - 17iT^{2} \) |
| 19 | \( 1 + 2iT - 19T^{2} \) |
| 23 | \( 1 + (-3.05 + 0.818i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 + (-2.23 - 3.87i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (2 - 3.46i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (3 + 3i)T + 37iT^{2} \) |
| 41 | \( 1 + (-7.74 - 4.47i)T + (20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-4.09 - 1.09i)T + (37.2 + 21.5i)T^{2} \) |
| 47 | \( 1 + (-9.16 - 2.45i)T + (40.7 + 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.23 - 2.23i)T + 53iT^{2} \) |
| 59 | \( 1 + (4.47 - 7.74i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (-3 - 5.19i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.36 + 0.366i)T + (58.0 - 33.5i)T^{2} \) |
| 71 | \( 1 + 4.47iT - 71T^{2} \) |
| 73 | \( 1 + (-1 + i)T - 73iT^{2} \) |
| 79 | \( 1 + (5.19 - 3i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-2.45 + 9.16i)T + (-71.8 - 41.5i)T^{2} \) |
| 89 | \( 1 - 4.47T + 89T^{2} \) |
| 97 | \( 1 + (-3.29 + 12.2i)T + (-84.0 - 48.5i)T^{2} \) |
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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
−9.237995329652230967273924526799, −8.775264110121731177753698826664, −7.67024955202400337378370013308, −7.11668929436845628409572955632, −6.17992152112189714236923857926, −5.47494615781897302292602987886, −4.28859737023943266536068504640, −3.23232694926330784702014635706, −2.69549955708055606099580761112, −0.887878267050731601774804034136,
0.957419846163250766956597661213, 2.05057720287685639847181117873, 3.83596827603476500592920920576, 4.11009323155347410691756056155, 5.12412047277643606535670835699, 6.19524567092682532375950480894, 7.08464891491273576553616859591, 7.69480317545725018765320932831, 8.709215091331595154985143389026, 9.315444924558336240346998680617