| L(s) = 1 | + (−1.21 − 1.59i)2-s + (−1.04 + 3.90i)3-s + (−1.06 + 3.85i)4-s + (4.34 − 2.48i)5-s + (7.48 − 3.06i)6-s + (−6.56 + 2.42i)7-s + (7.42 − 2.97i)8-s + (−6.36 − 3.67i)9-s + (−9.20 − 3.90i)10-s + (−4.38 + 2.52i)11-s + (−13.9 − 8.19i)12-s + (−14.0 + 14.0i)13-s + (11.8 + 7.50i)14-s + (5.14 + 19.5i)15-s + (−13.7 − 8.20i)16-s + (−5.60 + 20.9i)17-s + ⋯ |
| L(s) = 1 | + (−0.605 − 0.795i)2-s + (−0.348 + 1.30i)3-s + (−0.266 + 0.963i)4-s + (0.868 − 0.496i)5-s + (1.24 − 0.511i)6-s + (−0.938 + 0.346i)7-s + (0.928 − 0.372i)8-s + (−0.706 − 0.408i)9-s + (−0.920 − 0.390i)10-s + (−0.398 + 0.229i)11-s + (−1.16 − 0.682i)12-s + (−1.08 + 1.08i)13-s + (0.843 + 0.536i)14-s + (0.342 + 1.30i)15-s + (−0.858 − 0.512i)16-s + (−0.329 + 1.23i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.432 - 0.901i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.325271 + 0.516569i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.325271 + 0.516569i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.21 + 1.59i)T \) |
| 5 | \( 1 + (-4.34 + 2.48i)T \) |
| 7 | \( 1 + (6.56 - 2.42i)T \) |
| good | 3 | \( 1 + (1.04 - 3.90i)T + (-7.79 - 4.5i)T^{2} \) |
| 11 | \( 1 + (4.38 - 2.52i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (14.0 - 14.0i)T - 169iT^{2} \) |
| 17 | \( 1 + (5.60 - 20.9i)T + (-250. - 144.5i)T^{2} \) |
| 19 | \( 1 + (6.74 + 3.89i)T + (180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (3.35 + 12.5i)T + (-458. + 264.5i)T^{2} \) |
| 29 | \( 1 - 17.3iT - 841T^{2} \) |
| 31 | \( 1 + (-16.9 - 29.3i)T + (-480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (0.822 + 3.06i)T + (-1.18e3 + 684.5i)T^{2} \) |
| 41 | \( 1 + 59.4iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (3.18 - 3.18i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 + (-5.60 - 20.9i)T + (-1.91e3 + 1.10e3i)T^{2} \) |
| 53 | \( 1 + (19.3 - 72.1i)T + (-2.43e3 - 1.40e3i)T^{2} \) |
| 59 | \( 1 + (-77.5 + 44.7i)T + (1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-78.4 - 45.3i)T + (1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-22.2 + 83.0i)T + (-3.88e3 - 2.24e3i)T^{2} \) |
| 71 | \( 1 - 23.3iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-62.9 - 16.8i)T + (4.61e3 + 2.66e3i)T^{2} \) |
| 79 | \( 1 + (21.9 - 38.0i)T + (-3.12e3 - 5.40e3i)T^{2} \) |
| 83 | \( 1 + (-105. - 105. i)T + 6.88e3iT^{2} \) |
| 89 | \( 1 + (34.4 - 59.6i)T + (-3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (60.9 + 60.9i)T + 9.40e3iT^{2} \) |
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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
−12.82918244290538499975738440353, −12.24727738197273210721002285515, −10.78371419202012728472273381674, −10.10941196868819663231095564686, −9.415993472477497126920445386728, −8.668090765550724801564862798399, −6.70934211944068440934183803489, −5.12400064509225478147145609843, −4.03691503048135413856183885283, −2.31787367595533560336656146597,
0.48638149230374777875576570111, 2.46352201106810950083779305568, 5.36566217798806204052351862089, 6.34459615068713635694043477319, 7.09653880970332585488675138331, 7.943511702364397125124534136591, 9.616982727946034619242102334519, 10.17054492851617770037962322312, 11.56145381457907617061705378916, 13.11893769172041458474411623902
