| L(s) = 1 | − 2.74i·2-s − 3.53·4-s + (−2.32 − 1.34i)5-s + (0.122 − 6.99i)7-s − 1.27i·8-s + (−3.68 + 6.38i)10-s + (−7.18 + 4.14i)11-s + (1.91 + 3.31i)13-s + (−19.2 − 0.336i)14-s − 17.6·16-s + (−14.6 − 8.44i)17-s + (4.77 + 8.26i)19-s + (8.22 + 4.74i)20-s + (11.3 + 19.7i)22-s + (−21.1 − 12.2i)23-s + ⋯ |
| L(s) = 1 | − 1.37i·2-s − 0.883·4-s + (−0.465 − 0.268i)5-s + (0.0174 − 0.999i)7-s − 0.159i·8-s + (−0.368 + 0.638i)10-s + (−0.653 + 0.377i)11-s + (0.147 + 0.255i)13-s + (−1.37 − 0.0240i)14-s − 1.10·16-s + (−0.860 − 0.496i)17-s + (0.251 + 0.435i)19-s + (0.411 + 0.237i)20-s + (0.517 + 0.896i)22-s + (−0.920 − 0.531i)23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 189 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.921 - 0.389i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.201469 + 0.993649i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.201469 + 0.993649i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.122 + 6.99i)T \) |
| good | 2 | \( 1 + 2.74iT - 4T^{2} \) |
| 5 | \( 1 + (2.32 + 1.34i)T + (12.5 + 21.6i)T^{2} \) |
| 11 | \( 1 + (7.18 - 4.14i)T + (60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + (-1.91 - 3.31i)T + (-84.5 + 146. i)T^{2} \) |
| 17 | \( 1 + (14.6 + 8.44i)T + (144.5 + 250. i)T^{2} \) |
| 19 | \( 1 + (-4.77 - 8.26i)T + (-180.5 + 312. i)T^{2} \) |
| 23 | \( 1 + (21.1 + 12.2i)T + (264.5 + 458. i)T^{2} \) |
| 29 | \( 1 + (-41.1 - 23.7i)T + (420.5 + 728. i)T^{2} \) |
| 31 | \( 1 - 39.2T + 961T^{2} \) |
| 37 | \( 1 + (23.3 + 40.3i)T + (-684.5 + 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-51.2 + 29.5i)T + (840.5 - 1.45e3i)T^{2} \) |
| 43 | \( 1 + (-28.0 + 48.5i)T + (-924.5 - 1.60e3i)T^{2} \) |
| 47 | \( 1 + 21.5iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (22.2 + 12.8i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 - 46.3iT - 3.48e3T^{2} \) |
| 61 | \( 1 - 60.3T + 3.72e3T^{2} \) |
| 67 | \( 1 + 64.0T + 4.48e3T^{2} \) |
| 71 | \( 1 + 49.6iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-58.7 + 101. i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 - 40.5T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-64.7 - 37.3i)T + (3.44e3 + 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-59.9 + 34.6i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 + (46.9 - 81.3i)T + (-4.70e3 - 8.14e3i)T^{2} \) |
| 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
−11.87906139336460429458417433498, −10.65017203285190254716614027435, −10.29680158354626668889442663641, −9.044825791407987647497575296957, −7.80542710641963006343409850804, −6.65273449861376555024942556423, −4.67284759918867538957493478234, −3.83719834583362973670600787690, −2.32224115867471215654766035145, −0.58022597172565553430457814960,
2.70421916487595063734459663256, 4.61031569930190864077087312638, 5.79871619436872452370894944663, 6.55886351855613767688036230208, 7.928763366553977796117418904800, 8.355965304047601354128284435846, 9.610506528300878664042677596532, 11.06434079468233160661681864830, 11.86162116447302620457563107254, 13.16065211329843506013462829345
