| L(s) = 1 | + (−2.82 − 2.82i)2-s + 16.0i·4-s + (−50.1 + 50.1i)7-s + (45.2 − 45.2i)8-s + 659. i·11-s + (517. + 517. i)13-s + 283.·14-s − 256.·16-s + (62.7 + 62.7i)17-s − 38.1i·19-s + (1.86e3 − 1.86e3i)22-s + (−455. + 455. i)23-s − 2.92e3i·26-s + (−802. − 802. i)28-s − 4.59e3·29-s + ⋯ |
| L(s) = 1 | + (−0.499 − 0.499i)2-s + 0.500i·4-s + (−0.386 + 0.386i)7-s + (0.250 − 0.250i)8-s + 1.64i·11-s + (0.849 + 0.849i)13-s + 0.386·14-s − 0.250·16-s + (0.0526 + 0.0526i)17-s − 0.0242i·19-s + (0.821 − 0.821i)22-s + (−0.179 + 0.179i)23-s − 0.849i·26-s + (−0.193 − 0.193i)28-s − 1.01·29-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(6-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 450 ^{s/2} \, \Gamma_{\C}(s+5/2) \, L(s)\cr =\mathstrut & (-0.812 - 0.583i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(3)\) |
\(\approx\) |
\(0.7170779439\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7170779439\) |
| \(L(\frac{7}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (2.82 + 2.82i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + (50.1 - 50.1i)T - 1.68e4iT^{2} \) |
| 11 | \( 1 - 659. iT - 1.61e5T^{2} \) |
| 13 | \( 1 + (-517. - 517. i)T + 3.71e5iT^{2} \) |
| 17 | \( 1 + (-62.7 - 62.7i)T + 1.41e6iT^{2} \) |
| 19 | \( 1 + 38.1iT - 2.47e6T^{2} \) |
| 23 | \( 1 + (455. - 455. i)T - 6.43e6iT^{2} \) |
| 29 | \( 1 + 4.59e3T + 2.05e7T^{2} \) |
| 31 | \( 1 - 2.69e3T + 2.86e7T^{2} \) |
| 37 | \( 1 + (-1.48e3 + 1.48e3i)T - 6.93e7iT^{2} \) |
| 41 | \( 1 - 194. iT - 1.15e8T^{2} \) |
| 43 | \( 1 + (-9.86e3 - 9.86e3i)T + 1.47e8iT^{2} \) |
| 47 | \( 1 + (5.30e3 + 5.30e3i)T + 2.29e8iT^{2} \) |
| 53 | \( 1 + (-2.05e4 + 2.05e4i)T - 4.18e8iT^{2} \) |
| 59 | \( 1 + 1.86e4T + 7.14e8T^{2} \) |
| 61 | \( 1 + 4.48e4T + 8.44e8T^{2} \) |
| 67 | \( 1 + (-2.57e4 + 2.57e4i)T - 1.35e9iT^{2} \) |
| 71 | \( 1 + 6.14e4iT - 1.80e9T^{2} \) |
| 73 | \( 1 + (-2.72e4 - 2.72e4i)T + 2.07e9iT^{2} \) |
| 79 | \( 1 + 7.33e4iT - 3.07e9T^{2} \) |
| 83 | \( 1 + (5.89e4 - 5.89e4i)T - 3.93e9iT^{2} \) |
| 89 | \( 1 + 1.12e5T + 5.58e9T^{2} \) |
| 97 | \( 1 + (8.79e4 - 8.79e4i)T - 8.58e9iT^{2} \) |
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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
−10.59469567192373816549474037264, −9.581429358671094956762144005451, −9.181504766395696494550483705328, −7.999105921722239567630826199307, −7.08008713308660331425300621297, −6.12478670964153684932046150207, −4.70299367898076428438222680698, −3.72430117708553000618807934320, −2.37015800684098690127488459829, −1.44328799172644842837334907521,
0.22467282501031599211427066684, 1.14056479340578642001246083981, 2.93499127007061694842144289172, 3.96785188019091446138736717021, 5.56897540355783064099139562163, 6.10128229335484340488667579514, 7.22292157526806293431847144859, 8.225053476251629738402170367693, 8.809012663352016911087369441367, 9.899646246726798223474920220576
