| L(s) = 1 | + (−0.913 + 0.822i)2-s + (1.69 + 0.356i)3-s + (−0.0511 + 0.486i)4-s + (1.84 + 1.26i)5-s + (−1.84 + 1.06i)6-s + (1.60 + 0.925i)7-s + (−1.79 − 2.47i)8-s + (2.74 + 1.20i)9-s + (−2.72 + 0.360i)10-s + (−3.05 − 3.38i)11-s + (−0.260 + 0.806i)12-s + (−0.479 − 0.432i)13-s + (−2.22 + 0.472i)14-s + (2.67 + 2.80i)15-s + (2.72 + 0.578i)16-s + (0.197 + 0.271i)17-s + ⋯ |
| L(s) = 1 | + (−0.645 + 0.581i)2-s + (0.978 + 0.205i)3-s + (−0.0255 + 0.243i)4-s + (0.824 + 0.565i)5-s + (−0.751 + 0.436i)6-s + (0.605 + 0.349i)7-s + (−0.635 − 0.875i)8-s + (0.915 + 0.402i)9-s + (−0.861 + 0.114i)10-s + (−0.920 − 1.02i)11-s + (−0.0751 + 0.232i)12-s + (−0.133 − 0.119i)13-s + (−0.594 + 0.126i)14-s + (0.690 + 0.723i)15-s + (0.680 + 0.144i)16-s + (0.0478 + 0.0658i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.0745 - 0.997i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 225 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.0745 - 0.997i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.968210 + 0.898564i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.968210 + 0.898564i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (-1.69 - 0.356i)T \) |
| 5 | \( 1 + (-1.84 - 1.26i)T \) |
| good | 2 | \( 1 + (0.913 - 0.822i)T + (0.209 - 1.98i)T^{2} \) |
| 7 | \( 1 + (-1.60 - 0.925i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (3.05 + 3.38i)T + (-1.14 + 10.9i)T^{2} \) |
| 13 | \( 1 + (0.479 + 0.432i)T + (1.35 + 12.9i)T^{2} \) |
| 17 | \( 1 + (-0.197 - 0.271i)T + (-5.25 + 16.1i)T^{2} \) |
| 19 | \( 1 + (0.793 - 0.576i)T + (5.87 - 18.0i)T^{2} \) |
| 23 | \( 1 + (0.934 + 4.39i)T + (-21.0 + 9.35i)T^{2} \) |
| 29 | \( 1 + (8.49 - 3.78i)T + (19.4 - 21.5i)T^{2} \) |
| 31 | \( 1 + (1.46 + 0.652i)T + (20.7 + 23.0i)T^{2} \) |
| 37 | \( 1 + (-11.0 + 3.59i)T + (29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (4.98 - 5.53i)T + (-4.28 - 40.7i)T^{2} \) |
| 43 | \( 1 + (2.66 + 1.53i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + (1.02 + 2.31i)T + (-31.4 + 34.9i)T^{2} \) |
| 53 | \( 1 + (-6.39 + 8.80i)T + (-16.3 - 50.4i)T^{2} \) |
| 59 | \( 1 + (-8.68 + 9.64i)T + (-6.16 - 58.6i)T^{2} \) |
| 61 | \( 1 + (2.99 + 3.32i)T + (-6.37 + 60.6i)T^{2} \) |
| 67 | \( 1 + (1.54 - 3.47i)T + (-44.8 - 49.7i)T^{2} \) |
| 71 | \( 1 + (-2.38 - 1.73i)T + (21.9 + 67.5i)T^{2} \) |
| 73 | \( 1 + (2.05 + 0.667i)T + (59.0 + 42.9i)T^{2} \) |
| 79 | \( 1 + (2.59 - 1.15i)T + (52.8 - 58.7i)T^{2} \) |
| 83 | \( 1 + (-13.0 + 1.36i)T + (81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (2.56 - 7.89i)T + (-72.0 - 52.3i)T^{2} \) |
| 97 | \( 1 + (2.97 + 6.67i)T + (-64.9 + 72.0i)T^{2} \) |
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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
−12.87613109352382334448267416303, −11.24639748349696291511000952822, −10.22794766793729305441160222204, −9.337158206533404211652792067411, −8.422369898140514117285931334716, −7.81051977662037080959592837051, −6.65696936298354715646813126154, −5.36076611655141765604663696376, −3.54044372411253265054810251838, −2.35799490149306219759435927770,
1.54008979253143121742990048578, 2.46762928243745522694445126543, 4.50191896489541733186162724268, 5.68997008136939586206211661385, 7.37306633457641987567273210775, 8.264307113643508531344073806808, 9.362128235953949316855635444321, 9.811810097044030433584219637035, 10.74189334375005579194084849452, 11.99374594189804932836719164129
