| L(s) = 1 | + (−1.48 − 1.48i)2-s + (−2.63 + 2.63i)3-s − 11.5i·4-s + 7.83·6-s + (−13.0 − 13.0i)7-s + (−40.9 + 40.9i)8-s + 67.0i·9-s + 18.0·11-s + (30.6 + 30.6i)12-s + (1.78 − 1.78i)13-s + 38.8i·14-s − 64.0·16-s + (−8.06 − 8.06i)17-s + (99.5 − 99.5i)18-s + 201. i·19-s + ⋯ |
| L(s) = 1 | + (−0.370 − 0.370i)2-s + (−0.293 + 0.293i)3-s − 0.724i·4-s + 0.217·6-s + (−0.267 − 0.267i)7-s + (−0.639 + 0.639i)8-s + 0.828i·9-s + 0.149·11-s + (0.212 + 0.212i)12-s + (0.0105 − 0.0105i)13-s + 0.198i·14-s − 0.250·16-s + (−0.0278 − 0.0278i)17-s + (0.307 − 0.307i)18-s + 0.557i·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(5-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 175 ^{s/2} \, \Gamma_{\C}(s+2) \, L(s)\cr =\mathstrut & (0.973 - 0.229i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{5}{2})\) |
\(\approx\) |
\(1.107015067\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.107015067\) |
| \(L(3)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 7 | \( 1 + (13.0 + 13.0i)T \) |
| good | 2 | \( 1 + (1.48 + 1.48i)T + 16iT^{2} \) |
| 3 | \( 1 + (2.63 - 2.63i)T - 81iT^{2} \) |
| 11 | \( 1 - 18.0T + 1.46e4T^{2} \) |
| 13 | \( 1 + (-1.78 + 1.78i)T - 2.85e4iT^{2} \) |
| 17 | \( 1 + (8.06 + 8.06i)T + 8.35e4iT^{2} \) |
| 19 | \( 1 - 201. iT - 1.30e5T^{2} \) |
| 23 | \( 1 + (-414. + 414. i)T - 2.79e5iT^{2} \) |
| 29 | \( 1 - 1.06e3iT - 7.07e5T^{2} \) |
| 31 | \( 1 - 1.36e3T + 9.23e5T^{2} \) |
| 37 | \( 1 + (-848. - 848. i)T + 1.87e6iT^{2} \) |
| 41 | \( 1 - 2.80e3T + 2.82e6T^{2} \) |
| 43 | \( 1 + (322. - 322. i)T - 3.41e6iT^{2} \) |
| 47 | \( 1 + (-907. - 907. i)T + 4.87e6iT^{2} \) |
| 53 | \( 1 + (-3.08e3 + 3.08e3i)T - 7.89e6iT^{2} \) |
| 59 | \( 1 - 1.71e3iT - 1.21e7T^{2} \) |
| 61 | \( 1 + 4.95e3T + 1.38e7T^{2} \) |
| 67 | \( 1 + (-5.32e3 - 5.32e3i)T + 2.01e7iT^{2} \) |
| 71 | \( 1 - 3.84e3T + 2.54e7T^{2} \) |
| 73 | \( 1 + (3.29e3 - 3.29e3i)T - 2.83e7iT^{2} \) |
| 79 | \( 1 + 9.89e3iT - 3.89e7T^{2} \) |
| 83 | \( 1 + (5.25e3 - 5.25e3i)T - 4.74e7iT^{2} \) |
| 89 | \( 1 - 3.69e3iT - 6.27e7T^{2} \) |
| 97 | \( 1 + (-9.93e3 - 9.93e3i)T + 8.85e7iT^{2} \) |
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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
−11.78033053398538078885942729198, −10.81434510951229847988125587096, −10.27328821487731467765372670439, −9.284114054101168434233052198069, −8.154845871744334503439110889493, −6.70681465548285531112382439674, −5.56036472235821640818994876309, −4.48252421204211263124010387830, −2.61971520773009697426503820663, −1.03644868588281719050880874350,
0.61588207245870938811033514109, 2.81654189421668215735187937218, 4.13413258218981438637160880405, 5.91869588509886730712316268888, 6.81858466702621672116220536606, 7.78183460431559133202537827784, 8.971458640090334798360849700977, 9.633145837003006638474901502578, 11.24348801289311283789517074491, 12.06151578596619378555631496680
