| L(s) = 1 | + (−1.55 + 0.763i)3-s + (1.45 − 4.00i)5-s + (4.75 + 0.838i)7-s + (1.83 − 2.37i)9-s + (2.07 − 0.755i)11-s + (−3.48 + 2.92i)13-s + (0.792 + 7.34i)15-s + (3.55 − 2.05i)17-s + (−2.36 − 1.36i)19-s + (−8.03 + 2.32i)21-s + (−0.0197 − 0.112i)23-s + (−10.1 − 8.48i)25-s + (−1.03 + 5.09i)27-s + (1.43 − 1.70i)29-s + (−0.121 + 0.0213i)31-s + ⋯ |
| L(s) = 1 | + (−0.897 + 0.440i)3-s + (0.652 − 1.79i)5-s + (1.79 + 0.316i)7-s + (0.611 − 0.791i)9-s + (0.626 − 0.227i)11-s + (−0.966 + 0.810i)13-s + (0.204 + 1.89i)15-s + (0.863 − 0.498i)17-s + (−0.543 − 0.313i)19-s + (−1.75 + 0.507i)21-s + (−0.00412 − 0.0233i)23-s + (−2.02 − 1.69i)25-s + (−0.200 + 0.979i)27-s + (0.265 − 0.316i)29-s + (−0.0217 + 0.00383i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 864 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.644 + 0.764i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.44403 - 0.671489i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.44403 - 0.671489i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + (1.55 - 0.763i)T \) |
| good | 5 | \( 1 + (-1.45 + 4.00i)T + (-3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-4.75 - 0.838i)T + (6.57 + 2.39i)T^{2} \) |
| 11 | \( 1 + (-2.07 + 0.755i)T + (8.42 - 7.07i)T^{2} \) |
| 13 | \( 1 + (3.48 - 2.92i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-3.55 + 2.05i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (2.36 + 1.36i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (0.0197 + 0.112i)T + (-21.6 + 7.86i)T^{2} \) |
| 29 | \( 1 + (-1.43 + 1.70i)T + (-5.03 - 28.5i)T^{2} \) |
| 31 | \( 1 + (0.121 - 0.0213i)T + (29.1 - 10.6i)T^{2} \) |
| 37 | \( 1 + (-0.983 - 1.70i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-1.82 - 2.17i)T + (-7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (0.293 + 0.806i)T + (-32.9 + 27.6i)T^{2} \) |
| 47 | \( 1 + (1.43 - 8.13i)T + (-44.1 - 16.0i)T^{2} \) |
| 53 | \( 1 + 1.59iT - 53T^{2} \) |
| 59 | \( 1 + (-3.78 - 1.37i)T + (45.1 + 37.9i)T^{2} \) |
| 61 | \( 1 + (-2.31 + 13.1i)T + (-57.3 - 20.8i)T^{2} \) |
| 67 | \( 1 + (-2.36 - 2.81i)T + (-11.6 + 65.9i)T^{2} \) |
| 71 | \( 1 + (5.13 + 8.89i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (-2.77 + 4.79i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (1.16 - 1.38i)T + (-13.7 - 77.7i)T^{2} \) |
| 83 | \( 1 + (4.96 + 4.16i)T + (14.4 + 81.7i)T^{2} \) |
| 89 | \( 1 + (2.98 + 1.72i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.73 + 1.35i)T + (74.3 - 62.3i)T^{2} \) |
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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
−9.856877278705360459802509245994, −9.272373209324215393807572210581, −8.543220041240641656092304745460, −7.64589171354613744212455212638, −6.27540740655935083055578631271, −5.33502088259949290170447207234, −4.80514344831069819776512932979, −4.30225397495578699583227494059, −1.93343388205353066575239259718, −0.994879719476407770155781575282,
1.51780881009576319814959356580, 2.47301451810085571506501955905, 4.02309759410773844352166216115, 5.24671469241644151399236417457, 5.89740540792631550681880999973, 6.99119199674546611525649824426, 7.42917152353112373240675080645, 8.279632635227044740852933481502, 9.980979898168614252606063885127, 10.37511065475162115413316433675
