| L(s) = 1 | + (1.17 − 1.26i)3-s + (0.191 + 1.08i)5-s + (2.62 + 2.20i)7-s + (−0.225 − 2.99i)9-s + (−0.0811 + 0.460i)11-s + (3.01 + 1.09i)13-s + (1.60 + 1.03i)15-s + (−0.185 − 0.320i)17-s + (−2.75 + 4.77i)19-s + (5.88 − 0.738i)21-s + (0.107 − 0.0900i)23-s + (3.55 − 1.29i)25-s + (−4.06 − 3.23i)27-s + (1.79 − 0.651i)29-s + (2.20 − 1.85i)31-s + ⋯ |
| L(s) = 1 | + (0.679 − 0.733i)3-s + (0.0855 + 0.485i)5-s + (0.991 + 0.831i)7-s + (−0.0752 − 0.997i)9-s + (−0.0244 + 0.138i)11-s + (0.836 + 0.304i)13-s + (0.414 + 0.267i)15-s + (−0.0449 − 0.0777i)17-s + (−0.632 + 1.09i)19-s + (1.28 − 0.161i)21-s + (0.0223 − 0.0187i)23-s + (0.711 − 0.258i)25-s + (−0.782 − 0.622i)27-s + (0.332 − 0.121i)29-s + (0.396 − 0.332i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 + 0.0325i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 + 0.0325i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(2.25616 - 0.0366989i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(2.25616 - 0.0366989i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (-1.17 + 1.26i)T \) |
| good | 5 | \( 1 + (-0.191 - 1.08i)T + (-4.69 + 1.71i)T^{2} \) |
| 7 | \( 1 + (-2.62 - 2.20i)T + (1.21 + 6.89i)T^{2} \) |
| 11 | \( 1 + (0.0811 - 0.460i)T + (-10.3 - 3.76i)T^{2} \) |
| 13 | \( 1 + (-3.01 - 1.09i)T + (9.95 + 8.35i)T^{2} \) |
| 17 | \( 1 + (0.185 + 0.320i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (2.75 - 4.77i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-0.107 + 0.0900i)T + (3.99 - 22.6i)T^{2} \) |
| 29 | \( 1 + (-1.79 + 0.651i)T + (22.2 - 18.6i)T^{2} \) |
| 31 | \( 1 + (-2.20 + 1.85i)T + (5.38 - 30.5i)T^{2} \) |
| 37 | \( 1 + (-2.32 - 4.02i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.00243 - 0.000887i)T + (31.4 + 26.3i)T^{2} \) |
| 43 | \( 1 + (1.18 - 6.69i)T + (-40.4 - 14.7i)T^{2} \) |
| 47 | \( 1 + (6.94 + 5.83i)T + (8.16 + 46.2i)T^{2} \) |
| 53 | \( 1 - 10.8T + 53T^{2} \) |
| 59 | \( 1 + (1.70 + 9.67i)T + (-55.4 + 20.1i)T^{2} \) |
| 61 | \( 1 + (8.40 + 7.05i)T + (10.5 + 60.0i)T^{2} \) |
| 67 | \( 1 + (3.98 + 1.44i)T + (51.3 + 43.0i)T^{2} \) |
| 71 | \( 1 + (-0.776 - 1.34i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-7.58 + 13.1i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-8.87 + 3.23i)T + (60.5 - 50.7i)T^{2} \) |
| 83 | \( 1 + (16.2 - 5.91i)T + (63.5 - 53.3i)T^{2} \) |
| 89 | \( 1 + (-6.38 + 11.0i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (0.597 - 3.38i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.11257568207097249953703926635, −9.087558346104284651786391989883, −8.316581011397816955009598264559, −7.88534488789306920580030375036, −6.63641778544675210990458237399, −6.08765098157520683811814250876, −4.80343596148210642099548204845, −3.55936723895879963272459152614, −2.42840543144477807918659587373, −1.51360259272361330590059476088,
1.24667639702672203047640510462, 2.71459736921662807235393775440, 3.97789993663794454171125525433, 4.63510540212464393827770313152, 5.50292883736332610809146373156, 6.90065153133167909374846575871, 7.86564595150506204334854792555, 8.608101115426751575581571120851, 9.106190062141955047384538547666, 10.32575539858001407576077956461
