L(s) = 1 | + (0.125 + 1.99i)2-s + (−3.96 + 0.500i)4-s + (−3.32 + 3.32i)5-s − 4.04·7-s + (−1.49 − 7.85i)8-s + (−7.05 − 6.22i)10-s + (−6.82 − 6.82i)11-s + (4.29 + 4.29i)13-s + (−0.506 − 8.06i)14-s + (15.4 − 3.97i)16-s − 30.1·17-s + (−19.7 + 19.7i)19-s + (11.5 − 14.8i)20-s + (12.7 − 14.4i)22-s + 28.2·23-s + ⋯ |
L(s) = 1 | + (0.0626 + 0.998i)2-s + (−0.992 + 0.125i)4-s + (−0.665 + 0.665i)5-s − 0.577·7-s + (−0.187 − 0.982i)8-s + (−0.705 − 0.622i)10-s + (−0.620 − 0.620i)11-s + (0.330 + 0.330i)13-s + (−0.0361 − 0.576i)14-s + (0.968 − 0.248i)16-s − 1.77·17-s + (−1.03 + 1.03i)19-s + (0.576 − 0.743i)20-s + (0.580 − 0.658i)22-s + 1.22·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 144 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.799 + 0.600i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(0.134608 - 0.403772i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.134608 - 0.403772i\) |
\(L(2)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.125 - 1.99i)T \) |
| 3 | \( 1 \) |
good | 5 | \( 1 + (3.32 - 3.32i)T - 25iT^{2} \) |
| 7 | \( 1 + 4.04T + 49T^{2} \) |
| 11 | \( 1 + (6.82 + 6.82i)T + 121iT^{2} \) |
| 13 | \( 1 + (-4.29 - 4.29i)T + 169iT^{2} \) |
| 17 | \( 1 + 30.1T + 289T^{2} \) |
| 19 | \( 1 + (19.7 - 19.7i)T - 361iT^{2} \) |
| 23 | \( 1 - 28.2T + 529T^{2} \) |
| 29 | \( 1 + (-21.3 - 21.3i)T + 841iT^{2} \) |
| 31 | \( 1 + 38.0iT - 961T^{2} \) |
| 37 | \( 1 + (42.8 - 42.8i)T - 1.36e3iT^{2} \) |
| 41 | \( 1 - 48.2iT - 1.68e3T^{2} \) |
| 43 | \( 1 + (-32.6 - 32.6i)T + 1.84e3iT^{2} \) |
| 47 | \( 1 + 15.8iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (-0.476 + 0.476i)T - 2.80e3iT^{2} \) |
| 59 | \( 1 + (9.97 + 9.97i)T + 3.48e3iT^{2} \) |
| 61 | \( 1 + (-37.9 - 37.9i)T + 3.72e3iT^{2} \) |
| 67 | \( 1 + (-20.0 + 20.0i)T - 4.48e3iT^{2} \) |
| 71 | \( 1 + 40.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + 30.8iT - 5.32e3T^{2} \) |
| 79 | \( 1 + 130. iT - 6.24e3T^{2} \) |
| 83 | \( 1 + (-2.26 + 2.26i)T - 6.88e3iT^{2} \) |
| 89 | \( 1 + 72.2iT - 7.92e3T^{2} \) |
| 97 | \( 1 + 112.T + 9.40e3T^{2} \) |
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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
−13.46594257750043330574705781575, −12.86454994420864002996349298724, −11.36614311110453062510149452872, −10.40119024899604175851984078611, −9.013933504717810751072804773983, −8.139335946670904217562553244442, −6.91572204326723053211699795222, −6.18397529816965881690424606921, −4.56435458000215151691236704003, −3.26014392114705490427152221697,
0.26512459208611921806128814013, 2.49071078637066401221928321429, 4.10837837550872020640314920001, 5.05135856991110003713158344368, 6.87193336410669702094728097777, 8.493592353112041771357344203437, 9.073006888365541010670787930834, 10.47272254368801208944807204469, 11.16450484398519243153680764185, 12.50368486429840415485058297547