| L(s) = 1 | + (−0.984 + 0.173i)2-s + (−1.34 + 1.09i)3-s + (0.939 − 0.342i)4-s + (−2.60 − 2.18i)5-s + (1.13 − 1.30i)6-s + (−0.615 + 2.57i)7-s + (−0.866 + 0.5i)8-s + (0.616 − 2.93i)9-s + (2.94 + 1.69i)10-s + (0.604 + 0.720i)11-s + (−0.890 + 1.48i)12-s + (−0.436 − 0.0769i)13-s + (0.159 − 2.64i)14-s + (5.88 + 0.0950i)15-s + (0.766 − 0.642i)16-s + (1.11 − 1.92i)17-s + ⋯ |
| L(s) = 1 | + (−0.696 + 0.122i)2-s + (−0.776 + 0.630i)3-s + (0.469 − 0.171i)4-s + (−1.16 − 0.976i)5-s + (0.463 − 0.534i)6-s + (−0.232 + 0.972i)7-s + (−0.306 + 0.176i)8-s + (0.205 − 0.978i)9-s + (0.929 + 0.536i)10-s + (0.182 + 0.217i)11-s + (−0.256 + 0.428i)12-s + (−0.121 − 0.0213i)13-s + (0.0426 − 0.705i)14-s + (1.51 + 0.0245i)15-s + (0.191 − 0.160i)16-s + (0.269 − 0.467i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.998 + 0.0506i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.998 + 0.0506i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.567637 - 0.0143752i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.567637 - 0.0143752i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.984 - 0.173i)T \) |
| 3 | \( 1 + (1.34 - 1.09i)T \) |
| 7 | \( 1 + (0.615 - 2.57i)T \) |
| good | 5 | \( 1 + (2.60 + 2.18i)T + (0.868 + 4.92i)T^{2} \) |
| 11 | \( 1 + (-0.604 - 0.720i)T + (-1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (0.436 + 0.0769i)T + (12.2 + 4.44i)T^{2} \) |
| 17 | \( 1 + (-1.11 + 1.92i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-6.85 + 3.95i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.171 + 0.470i)T + (-17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-8.00 + 1.41i)T + (27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-2.68 - 7.37i)T + (-23.7 + 19.9i)T^{2} \) |
| 37 | \( 1 + (0.250 - 0.433i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.16 + 6.61i)T + (-38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-3.79 + 3.18i)T + (7.46 - 42.3i)T^{2} \) |
| 47 | \( 1 + (-2.17 - 0.790i)T + (36.0 + 30.2i)T^{2} \) |
| 53 | \( 1 - 7.36iT - 53T^{2} \) |
| 59 | \( 1 + (0.00253 + 0.00213i)T + (10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-4.17 + 11.4i)T + (-46.7 - 39.2i)T^{2} \) |
| 67 | \( 1 + (-2.67 + 15.1i)T + (-62.9 - 22.9i)T^{2} \) |
| 71 | \( 1 + (10.3 + 5.98i)T + (35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (3.87 - 2.23i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.93 - 10.9i)T + (-74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (2.34 + 13.2i)T + (-77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (-4.96 - 8.60i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-1.02 - 1.21i)T + (-16.8 + 95.5i)T^{2} \) |
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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
−11.52228741377417921180164625066, −10.38748749364102563133863322569, −9.308677397109490324533364245178, −8.865942490326729980084072090865, −7.73959878186663676610150000268, −6.66745820924219414678427121881, −5.36768470878643686134665886584, −4.66907693675858801168230137189, −3.17440595578831303048754252527, −0.74662676181923406948834428051,
0.978728492214556647775958053069, 3.03114052726543404096868909002, 4.20788842087699793288070399113, 5.95063810854537832898096061188, 6.93961152575041226891214237642, 7.55392645983175156078890699719, 8.183905213509800035190585441424, 9.962504546527451989580905026273, 10.46455888329863619894552848165, 11.56730175065106944676635523744
