L(s) = 1 | + (0.173 − 0.984i)2-s + (−0.390 − 1.68i)3-s + (−0.939 − 0.342i)4-s + (0.393 − 0.329i)5-s + (−1.72 + 0.0916i)6-s + (−0.939 + 0.342i)7-s + (−0.5 + 0.866i)8-s + (−2.69 + 1.31i)9-s + (−0.256 − 0.444i)10-s + (−2.36 − 1.98i)11-s + (−0.210 + 1.71i)12-s + (−0.932 − 5.29i)13-s + (0.173 + 0.984i)14-s + (−0.710 − 0.534i)15-s + (0.766 + 0.642i)16-s + (−2.52 − 4.37i)17-s + ⋯ |
L(s) = 1 | + (0.122 − 0.696i)2-s + (−0.225 − 0.974i)3-s + (−0.469 − 0.171i)4-s + (0.175 − 0.147i)5-s + (−0.706 + 0.0374i)6-s + (−0.355 + 0.129i)7-s + (−0.176 + 0.306i)8-s + (−0.898 + 0.439i)9-s + (−0.0811 − 0.140i)10-s + (−0.714 − 0.599i)11-s + (−0.0606 + 0.496i)12-s + (−0.258 − 1.46i)13-s + (0.0464 + 0.263i)14-s + (−0.183 − 0.138i)15-s + (0.191 + 0.160i)16-s + (−0.612 − 1.06i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.969 - 0.245i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.103369 + 0.828018i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.103369 + 0.828018i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.173 + 0.984i)T \) |
| 3 | \( 1 + (0.390 + 1.68i)T \) |
| 7 | \( 1 + (0.939 - 0.342i)T \) |
good | 5 | \( 1 + (-0.393 + 0.329i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (2.36 + 1.98i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.932 + 5.29i)T + (-12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (2.52 + 4.37i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.98 - 5.16i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.07 - 1.48i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.663 + 3.76i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-7.88 - 2.87i)T + (23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (3.24 + 5.62i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (1.73 + 9.83i)T + (-38.5 + 14.0i)T^{2} \) |
| 43 | \( 1 + (5.49 + 4.61i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-12.7 + 4.63i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 2.76T + 53T^{2} \) |
| 59 | \( 1 + (-4.66 + 3.91i)T + (10.2 - 58.1i)T^{2} \) |
| 61 | \( 1 + (5.17 - 1.88i)T + (46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.300 - 1.70i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (1.26 + 2.19i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-4.87 + 8.44i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.55 - 8.80i)T + (-74.2 - 27.0i)T^{2} \) |
| 83 | \( 1 + (-1.50 + 8.54i)T + (-77.9 - 28.3i)T^{2} \) |
| 89 | \( 1 + (0.964 - 1.67i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-11.3 - 9.56i)T + (16.8 + 95.5i)T^{2} \) |
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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
−10.87695834758595366614126655534, −10.26962840145302206369132337823, −8.975780180123411042713595599723, −8.128504420678846553840631675589, −7.13059579111580261683113943337, −5.80640328282853040886654389017, −5.18863090811921907563806946024, −3.31700702797424029257380303677, −2.29588860622682684348711465207, −0.53240959774674558491690338395,
2.70370745430852939930921421962, 4.34534857719467043465416942173, 4.77546741806179970922168179829, 6.28532245773757685968777533666, 6.80043736353107858892681371283, 8.305209431984605964589918634725, 9.103888619874867956338295466804, 9.975131405931042496855395555117, 10.75431418488516430273979391405, 11.77012770376185357229487762850