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
−11.77012770376185357229487762850, −10.75431418488516430273979391405, −9.975131405931042496855395555117, −9.103888619874867956338295466804, −8.305209431984605964589918634725, −6.80043736353107858892681371283, −6.28532245773757685968777533666, −4.77546741806179970922168179829, −4.34534857719467043465416942173, −2.70370745430852939930921421962,
0.53240959774674558491690338395, 2.29588860622682684348711465207, 3.31700702797424029257380303677, 5.18863090811921907563806946024, 5.80640328282853040886654389017, 7.13059579111580261683113943337, 8.128504420678846553840631675589, 8.975780180123411042713595599723, 10.26962840145302206369132337823, 10.87695834758595366614126655534