| L(s) = 1 | + (0.939 − 0.342i)2-s + (1.69 − 0.356i)3-s + (0.766 − 0.642i)4-s + (−0.642 + 0.538i)5-s + (1.47 − 0.914i)6-s + (1.56 − 2.12i)7-s + (0.500 − 0.866i)8-s + (2.74 − 1.20i)9-s + (−0.419 + 0.725i)10-s + (−2.11 − 1.77i)11-s + (1.06 − 1.36i)12-s + (−4.68 + 3.92i)13-s + (0.746 − 2.53i)14-s + (−0.896 + 1.14i)15-s + (0.173 − 0.984i)16-s + (0.609 − 1.05i)17-s + ⋯ |
| L(s) = 1 | + (0.664 − 0.241i)2-s + (0.978 − 0.205i)3-s + (0.383 − 0.321i)4-s + (−0.287 + 0.240i)5-s + (0.600 − 0.373i)6-s + (0.593 − 0.804i)7-s + (0.176 − 0.306i)8-s + (0.915 − 0.402i)9-s + (−0.132 + 0.229i)10-s + (−0.637 − 0.535i)11-s + (0.308 − 0.393i)12-s + (−1.29 + 1.08i)13-s + (0.199 − 0.678i)14-s + (−0.231 + 0.294i)15-s + (0.0434 − 0.246i)16-s + (0.147 − 0.256i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 378 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.760 + 0.649i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.40452 - 0.886466i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.40452 - 0.886466i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (-0.939 + 0.342i)T \) |
| 3 | \( 1 + (-1.69 + 0.356i)T \) |
| 7 | \( 1 + (-1.56 + 2.12i)T \) |
| good | 5 | \( 1 + (0.642 - 0.538i)T + (0.868 - 4.92i)T^{2} \) |
| 11 | \( 1 + (2.11 + 1.77i)T + (1.91 + 10.8i)T^{2} \) |
| 13 | \( 1 + (4.68 - 3.92i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.609 + 1.05i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.80 - 6.59i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.52 + 0.553i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (0.112 + 0.0943i)T + (5.03 + 28.5i)T^{2} \) |
| 31 | \( 1 + (-6.35 + 5.33i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + 0.0134T + 37T^{2} \) |
| 41 | \( 1 + (8.78 - 7.37i)T + (7.11 - 40.3i)T^{2} \) |
| 43 | \( 1 + (5.97 - 2.17i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (8.41 + 7.06i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 + (0.294 + 0.509i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-2.08 - 11.8i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (-3.77 - 3.17i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (-1.68 - 0.612i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (5.50 + 9.53i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 - 5.06T + 73T^{2} \) |
| 79 | \( 1 + (0.117 - 0.0428i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (-8.76 - 7.35i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.46 + 4.26i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-8.45 + 3.07i)T + (74.3 - 62.3i)T^{2} \) |
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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
−11.56710610152295608098880174693, −10.20705899292887518346957862248, −9.710719713387019723119821634100, −8.177290535027833116754293927949, −7.58095819892571709775803685894, −6.67735072752966120318341575194, −5.10414252986075683960124307970, −4.07380641290321562548172641739, −3.07342445244392436852503665600, −1.71237470409140448086170284208,
2.28300316691701711940610715950, 3.16743569834726578765357023694, 4.82153433329629577009010132292, 5.11635220518307677507368451571, 6.89727756853377316668752291237, 7.87120847064715354863365862492, 8.397924832889030774269896102019, 9.614285137292675940601844973213, 10.44834266330669883471033486239, 11.78139546441188291830544793160
