L(s) = 1 | + (−23.2 − 23.2i)3-s + (41.0 + 41.0i)5-s − 327.·7-s + 354. i·9-s + (266. − 266. i)11-s + (1.20e3 − 1.20e3i)13-s − 1.91e3i·15-s + 36.0·17-s + (−7.73e3 − 7.73e3i)19-s + (7.61e3 + 7.61e3i)21-s − 8.60e3·23-s − 1.22e4i·25-s + (−8.72e3 + 8.72e3i)27-s + (−1.92e4 + 1.92e4i)29-s − 56.8i·31-s + ⋯ |
L(s) = 1 | + (−0.861 − 0.861i)3-s + (0.328 + 0.328i)5-s − 0.953·7-s + 0.485i·9-s + (0.199 − 0.199i)11-s + (0.546 − 0.546i)13-s − 0.565i·15-s + 0.00732·17-s + (−1.12 − 1.12i)19-s + (0.821 + 0.821i)21-s − 0.707·23-s − 0.784i·25-s + (−0.443 + 0.443i)27-s + (−0.791 + 0.791i)29-s − 0.00190i·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(7-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 256 ^{s/2} \, \Gamma_{\C}(s+3) \, L(s)\cr =\mathstrut & (0.382 - 0.923i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{7}{2})\) |
\(\approx\) |
\(0.3886908942\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.3886908942\) |
\(L(4)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
good | 3 | \( 1 + (23.2 + 23.2i)T + 729iT^{2} \) |
| 5 | \( 1 + (-41.0 - 41.0i)T + 1.56e4iT^{2} \) |
| 7 | \( 1 + 327.T + 1.17e5T^{2} \) |
| 11 | \( 1 + (-266. + 266. i)T - 1.77e6iT^{2} \) |
| 13 | \( 1 + (-1.20e3 + 1.20e3i)T - 4.82e6iT^{2} \) |
| 17 | \( 1 - 36.0T + 2.41e7T^{2} \) |
| 19 | \( 1 + (7.73e3 + 7.73e3i)T + 4.70e7iT^{2} \) |
| 23 | \( 1 + 8.60e3T + 1.48e8T^{2} \) |
| 29 | \( 1 + (1.92e4 - 1.92e4i)T - 5.94e8iT^{2} \) |
| 31 | \( 1 + 56.8iT - 8.87e8T^{2} \) |
| 37 | \( 1 + (-6.12e4 - 6.12e4i)T + 2.56e9iT^{2} \) |
| 41 | \( 1 + 8.59e4iT - 4.75e9T^{2} \) |
| 43 | \( 1 + (5.85e4 - 5.85e4i)T - 6.32e9iT^{2} \) |
| 47 | \( 1 + 1.57e5iT - 1.07e10T^{2} \) |
| 53 | \( 1 + (1.99e4 + 1.99e4i)T + 2.21e10iT^{2} \) |
| 59 | \( 1 + (1.86e4 - 1.86e4i)T - 4.21e10iT^{2} \) |
| 61 | \( 1 + (2.65e3 - 2.65e3i)T - 5.15e10iT^{2} \) |
| 67 | \( 1 + (-1.18e5 - 1.18e5i)T + 9.04e10iT^{2} \) |
| 71 | \( 1 - 5.22e5T + 1.28e11T^{2} \) |
| 73 | \( 1 - 4.32e5iT - 1.51e11T^{2} \) |
| 79 | \( 1 + 3.74e4iT - 2.43e11T^{2} \) |
| 83 | \( 1 + (-7.55e5 - 7.55e5i)T + 3.26e11iT^{2} \) |
| 89 | \( 1 - 1.19e6iT - 4.96e11T^{2} \) |
| 97 | \( 1 - 1.01e6T + 8.32e11T^{2} \) |
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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
−11.19257936230010305900602451070, −10.36321311213776868031120822146, −9.286029749531189932185142739727, −8.102230627549603360636340993508, −6.61960596133439460071173029841, −6.50962385933319860306241059349, −5.35510561709015973516210072934, −3.69706204572397923521908263880, −2.33575294589912364523461434928, −0.851256837838648088204577340358,
0.14160077039597802801782371510, 1.86497166787295644813104216114, 3.66949512329690149729880133778, 4.52054423501617541842775939902, 5.84663739467060513572315339861, 6.31845095694128136760591822509, 7.88125894827137508539123366773, 9.258170220221017209604556945681, 9.811875365831341230210835918069, 10.73997857072541705615599822874