| L(s) = 1 | + (−1.39 + 0.296i)2-s + (−0.399 − 1.68i)3-s + (0.0344 − 0.0153i)4-s + (−2.10 + 0.756i)5-s + (1.05 + 2.23i)6-s + (1.20 + 2.08i)7-s + (2.26 − 1.64i)8-s + (−2.68 + 1.34i)9-s + (2.71 − 1.68i)10-s + (0.0309 − 0.00656i)11-s + (−0.0395 − 0.0518i)12-s + (6.17 + 1.31i)13-s + (−2.30 − 2.55i)14-s + (2.11 + 3.24i)15-s + (−2.72 + 3.02i)16-s + (2.72 − 1.97i)17-s + ⋯ |
| L(s) = 1 | + (−0.987 + 0.209i)2-s + (−0.230 − 0.973i)3-s + (0.0172 − 0.00765i)4-s + (−0.941 + 0.338i)5-s + (0.432 + 0.912i)6-s + (0.455 + 0.788i)7-s + (0.801 − 0.582i)8-s + (−0.893 + 0.449i)9-s + (0.858 − 0.531i)10-s + (0.00931 − 0.00198i)11-s + (−0.0114 − 0.0149i)12-s + (1.71 + 0.363i)13-s + (−0.615 − 0.683i)14-s + (0.546 + 0.837i)15-s + (−0.681 + 0.756i)16-s + (0.659 − 0.479i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.909 - 0.416i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.557291 + 0.121551i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.557291 + 0.121551i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.399 + 1.68i)T \) |
| 5 | \( 1 + (2.10 - 0.756i)T \) |
| good | 2 | \( 1 + (1.39 - 0.296i)T + (1.82 - 0.813i)T^{2} \) |
| 7 | \( 1 + (-1.20 - 2.08i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.0309 + 0.00656i)T + (10.0 - 4.47i)T^{2} \) |
| 13 | \( 1 + (-6.17 - 1.31i)T + (11.8 + 5.28i)T^{2} \) |
| 17 | \( 1 + (-2.72 + 1.97i)T + (5.25 - 16.1i)T^{2} \) |
| 19 | \( 1 + (-1.81 + 1.31i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (-1.28 - 1.42i)T + (-2.40 + 22.8i)T^{2} \) |
| 29 | \( 1 + (0.977 - 9.29i)T + (-28.3 - 6.02i)T^{2} \) |
| 31 | \( 1 + (-0.393 - 3.73i)T + (-30.3 + 6.44i)T^{2} \) |
| 37 | \( 1 + (-3.22 - 9.94i)T + (-29.9 + 21.7i)T^{2} \) |
| 41 | \( 1 + (1.58 + 0.337i)T + (37.4 + 16.6i)T^{2} \) |
| 43 | \( 1 + (5.58 + 9.67i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (-0.330 + 3.14i)T + (-45.9 - 9.77i)T^{2} \) |
| 53 | \( 1 + (-1.23 - 0.896i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-12.5 - 2.66i)T + (53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (10.3 - 2.20i)T + (55.7 - 24.8i)T^{2} \) |
| 67 | \( 1 + (-0.237 - 2.26i)T + (-65.5 + 13.9i)T^{2} \) |
| 71 | \( 1 + (3.63 + 2.64i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (-0.380 + 1.17i)T + (-59.0 - 42.9i)T^{2} \) |
| 79 | \( 1 + (0.0961 - 0.915i)T + (-77.2 - 16.4i)T^{2} \) |
| 83 | \( 1 + (-5.07 - 2.26i)T + (55.5 + 61.6i)T^{2} \) |
| 89 | \( 1 + (0.932 - 2.86i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (-1.08 + 10.3i)T + (-94.8 - 20.1i)T^{2} \) |
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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
−12.04592133887050028244335411474, −11.41867307236631866694289705979, −10.48408613398839698776297096659, −8.792269363810061245128645468808, −8.503912054757202818151429300532, −7.44398289877251206049053279103, −6.65155098038879207514372809907, −5.16387513688833444801684930428, −3.38060194594487674637653900661, −1.27503814768059339124837710296,
0.868550875409427512505221783306, 3.69739657683625024996186145039, 4.50817977403360005720950256512, 5.86593442422373215870300941591, 7.77442574102206289771993658480, 8.293071551176849952870366082438, 9.302150686673749004943744727573, 10.28540848653170903589856289516, 11.05034297881298980827365749381, 11.56260174414187491515907115053
