L(s) = 1 | + (1.94 − 3.37i)3-s + (−4.42 − 7.67i)5-s + (6.92 + 1.03i)7-s + (−3.09 − 5.35i)9-s + (3.15 + 1.82i)11-s − 7.79·13-s − 34.5·15-s + (−9.07 − 5.23i)17-s + (−5.39 − 9.34i)19-s + (16.9 − 21.3i)21-s + (−6.45 − 11.1i)23-s + (−26.7 + 46.3i)25-s + 10.9·27-s + 17.2i·29-s + (26.1 + 15.1i)31-s + ⋯ |
L(s) = 1 | + (0.649 − 1.12i)3-s + (−0.885 − 1.53i)5-s + (0.989 + 0.147i)7-s + (−0.343 − 0.594i)9-s + (0.287 + 0.165i)11-s − 0.599·13-s − 2.30·15-s + (−0.533 − 0.308i)17-s + (−0.283 − 0.491i)19-s + (0.808 − 1.01i)21-s + (−0.280 − 0.486i)23-s + (−1.06 + 1.85i)25-s + 0.406·27-s + 0.594i·29-s + (0.844 + 0.487i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 224 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.657 + 0.753i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.685394 - 1.50795i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.685394 - 1.50795i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 7 | \( 1 + (-6.92 - 1.03i)T \) |
good | 3 | \( 1 + (-1.94 + 3.37i)T + (-4.5 - 7.79i)T^{2} \) |
| 5 | \( 1 + (4.42 + 7.67i)T + (-12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (-3.15 - 1.82i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + 7.79T + 169T^{2} \) |
| 17 | \( 1 + (9.07 + 5.23i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (5.39 + 9.34i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (6.45 + 11.1i)T + (-264.5 + 458. i)T^{2} \) |
| 29 | \( 1 - 17.2iT - 841T^{2} \) |
| 31 | \( 1 + (-26.1 - 15.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (-34.2 + 19.7i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 73.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 - 40.8iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-36.2 + 20.9i)T + (1.10e3 - 1.91e3i)T^{2} \) |
| 53 | \( 1 + (5.55 + 3.20i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (-7.95 + 13.7i)T + (-1.74e3 - 3.01e3i)T^{2} \) |
| 61 | \( 1 + (6.07 + 10.5i)T + (-1.86e3 + 3.22e3i)T^{2} \) |
| 67 | \( 1 + (6.75 + 3.89i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 41.3T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-77.6 - 44.8i)T + (2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-35.3 - 61.3i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 - 60.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + (23.4 - 13.5i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 3.26iT - 9.40e3T^{2} \) |
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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
−12.13445738266258683450997916150, −11.01347397798080996784429047701, −9.230645351090332748385058292202, −8.478949526925327871811557211051, −7.87531999231234154199544547201, −6.94578856257837151606083756442, −5.14176450883272572540563182866, −4.26787163026277701201008303932, −2.22296676223206936382118416601, −0.888892850848509171333454312569,
2.59407949589186077315603512911, 3.78650174445159625533375802946, 4.54334041450688068671045349957, 6.35662765581023485373847832234, 7.61281743312980869417888657292, 8.312098468557966139808422884776, 9.653686078977874116872648616657, 10.47821745352505864976345231075, 11.22356807371583018159381018337, 11.97693549299699794557168384871