L(s) = 1 | + (−1.13 + 2.77i)3-s + (−6.28 − 6.28i)5-s − 1.64i·7-s + (−6.43 − 6.29i)9-s + (4.75 + 4.75i)11-s + (9.35 + 9.35i)13-s + (24.5 − 10.3i)15-s + 11.4i·17-s + (8.58 + 8.58i)19-s + (4.57 + 1.86i)21-s + 16.2·23-s + 54.0i·25-s + (24.7 − 10.7i)27-s + (10.7 − 10.7i)29-s + 6.35·31-s + ⋯ |
L(s) = 1 | + (−0.377 + 0.926i)3-s + (−1.25 − 1.25i)5-s − 0.235i·7-s + (−0.715 − 0.699i)9-s + (0.432 + 0.432i)11-s + (0.719 + 0.719i)13-s + (1.63 − 0.689i)15-s + 0.675i·17-s + (0.451 + 0.451i)19-s + (0.217 + 0.0887i)21-s + 0.706·23-s + 2.16i·25-s + (0.917 − 0.398i)27-s + (0.370 − 0.370i)29-s + 0.204·31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 384 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.537 - 0.843i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.919712 + 0.504278i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.919712 + 0.504278i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.13 - 2.77i)T \) |
good | 5 | \( 1 + (6.28 + 6.28i)T + 25iT^{2} \) |
| 7 | \( 1 + 1.64iT - 49T^{2} \) |
| 11 | \( 1 + (-4.75 - 4.75i)T + 121iT^{2} \) |
| 13 | \( 1 + (-9.35 - 9.35i)T + 169iT^{2} \) |
| 17 | \( 1 - 11.4iT - 289T^{2} \) |
| 19 | \( 1 + (-8.58 - 8.58i)T + 361iT^{2} \) |
| 23 | \( 1 - 16.2T + 529T^{2} \) |
| 29 | \( 1 + (-10.7 + 10.7i)T - 841iT^{2} \) |
| 31 | \( 1 - 6.35T + 961T^{2} \) |
| 37 | \( 1 + (27.2 - 27.2i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 1.98T + 1.68e3T^{2} \) |
| 43 | \( 1 + (-19.4 + 19.4i)T - 1.84e3iT^{2} \) |
| 47 | \( 1 - 74.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (4.00 + 4.00i)T + 2.80e3iT^{2} \) |
| 59 | \( 1 + (-27.9 - 27.9i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (39.2 + 39.2i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-68.6 - 68.6i)T + 4.48e3iT^{2} \) |
| 71 | \( 1 - 40.6T + 5.04e3T^{2} \) |
| 73 | \( 1 + 59.0iT - 5.32e3T^{2} \) |
| 79 | \( 1 - 17.3T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-75.1 + 75.1i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 - 78.8T + 7.92e3T^{2} \) |
| 97 | \( 1 + 38.8T + 9.40e3T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−11.40106353445426654713521843595, −10.43322656880462054281939187548, −9.265646592630187630031939550627, −8.703186612646742555133663702715, −7.72993879023578696724674767256, −6.39732115801581149005806582425, −5.10809955894694567539289982681, −4.27117322926042117478366909885, −3.62438303496762471810317013131, −1.05332531094639369396394099107,
0.64477962522212284094104441036, 2.69974637061842803290563608221, 3.62220533689448611069439870348, 5.28764567486036066548585617106, 6.49322336763243039880901346998, 7.14393015627368323265434041810, 7.953661522454681265928693248080, 8.861377385398031266959357090517, 10.50477394482935578780768306442, 11.16882358685399179042054208853