L(s) = 1 | + (0.998 − 1.00i)2-s + (−0.605 − 1.62i)3-s + (−0.00424 − 1.99i)4-s + 0.171i·5-s + (−2.22 − 1.01i)6-s + (2.51 + 1.45i)7-s + (−2.00 − 1.99i)8-s + (−2.26 + 1.96i)9-s + (0.171 + 0.171i)10-s + (−0.674 − 1.16i)11-s + (−3.24 + 1.21i)12-s + (−3.29 − 1.45i)13-s + (3.97 − 1.06i)14-s + (0.278 − 0.103i)15-s + (−3.99 + 0.0169i)16-s + (4.48 + 2.59i)17-s + ⋯ |
L(s) = 1 | + (0.706 − 0.707i)2-s + (−0.349 − 0.936i)3-s + (−0.00212 − 0.999i)4-s + 0.0767i·5-s + (−0.910 − 0.414i)6-s + (0.952 + 0.549i)7-s + (−0.709 − 0.704i)8-s + (−0.755 + 0.654i)9-s + (0.0543 + 0.0541i)10-s + (−0.203 − 0.352i)11-s + (−0.936 + 0.351i)12-s + (−0.915 − 0.403i)13-s + (1.06 − 0.285i)14-s + (0.0718 − 0.0268i)15-s + (−0.999 + 0.00424i)16-s + (1.08 + 0.628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 156 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.306 + 0.951i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.857559 - 1.17679i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.857559 - 1.17679i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.998 + 1.00i)T \) |
| 3 | \( 1 + (0.605 + 1.62i)T \) |
| 13 | \( 1 + (3.29 + 1.45i)T \) |
good | 5 | \( 1 - 0.171iT - 5T^{2} \) |
| 7 | \( 1 + (-2.51 - 1.45i)T + (3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (0.674 + 1.16i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-4.48 - 2.59i)T + (8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.90 - 2.82i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.40 - 2.44i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (2.51 - 1.45i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 3.62iT - 31T^{2} \) |
| 37 | \( 1 + (1.64 + 2.85i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (4.85 - 2.80i)T + (20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (6.79 + 3.92i)T + (21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 7.05T + 47T^{2} \) |
| 53 | \( 1 - 12.1iT - 53T^{2} \) |
| 59 | \( 1 + (-4.93 + 8.55i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (0.259 - 0.449i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (9.87 - 5.70i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (5.79 - 10.0i)T + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 - 1.09T + 73T^{2} \) |
| 79 | \( 1 - 5.99iT - 79T^{2} \) |
| 83 | \( 1 - 3.19T + 83T^{2} \) |
| 89 | \( 1 + (-3.87 + 2.23i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (4.06 - 7.03i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−12.46096693258718623262306744087, −11.85622260805605169665685913055, −11.02573803419200003884444280408, −9.908577538536051463262109470569, −8.366764554396340686265174261489, −7.27073735396350240597651518708, −5.71895967816473012959995120989, −5.13208153645232546012354306316, −3.07011103497372060138822302969, −1.54866222805574192835962944361,
3.21868339439269603445367691856, 4.83147903673103981449008048290, 5.07320570719844894358772591243, 6.82367616879017494190580891608, 7.82888743541136071329595295282, 9.103675028189436145597944811581, 10.24443886028015167999060542505, 11.50385590465746744553018093399, 12.09006096536857838419071712508, 13.49250710065626685080046739180