| L(s) = 1 | + (−2.56 + 0.607i)2-s + (−0.0414 + 1.73i)3-s + (4.40 − 2.21i)4-s + (0.734 + 0.987i)5-s + (−0.945 − 4.46i)6-s + (−3.16 + 2.08i)7-s + (−5.90 + 4.95i)8-s + (−2.99 − 0.143i)9-s + (−2.48 − 2.08i)10-s + (0.129 + 0.0151i)11-s + (3.64 + 7.71i)12-s + (0.399 + 1.33i)13-s + (6.84 − 7.25i)14-s + (−1.73 + 1.23i)15-s + (6.23 − 8.37i)16-s + (0.456 − 2.59i)17-s + ⋯ |
| L(s) = 1 | + (−1.81 + 0.429i)2-s + (−0.0239 + 0.999i)3-s + (2.20 − 1.10i)4-s + (0.328 + 0.441i)5-s + (−0.385 − 1.82i)6-s + (−1.19 + 0.786i)7-s + (−2.08 + 1.75i)8-s + (−0.998 − 0.0478i)9-s + (−0.784 − 0.658i)10-s + (0.0391 + 0.00457i)11-s + (1.05 + 2.22i)12-s + (0.110 + 0.369i)13-s + (1.82 − 1.93i)14-s + (−0.449 + 0.318i)15-s + (1.55 − 2.09i)16-s + (0.110 − 0.628i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 81 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.754 - 0.656i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.132563 + 0.354198i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.132563 + 0.354198i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (0.0414 - 1.73i)T \) |
| good | 2 | \( 1 + (2.56 - 0.607i)T + (1.78 - 0.897i)T^{2} \) |
| 5 | \( 1 + (-0.734 - 0.987i)T + (-1.43 + 4.78i)T^{2} \) |
| 7 | \( 1 + (3.16 - 2.08i)T + (2.77 - 6.42i)T^{2} \) |
| 11 | \( 1 + (-0.129 - 0.0151i)T + (10.7 + 2.53i)T^{2} \) |
| 13 | \( 1 + (-0.399 - 1.33i)T + (-10.8 + 7.14i)T^{2} \) |
| 17 | \( 1 + (-0.456 + 2.59i)T + (-15.9 - 5.81i)T^{2} \) |
| 19 | \( 1 + (-0.985 - 5.58i)T + (-17.8 + 6.49i)T^{2} \) |
| 23 | \( 1 + (-2.62 - 1.72i)T + (9.10 + 21.1i)T^{2} \) |
| 29 | \( 1 + (-6.43 - 6.82i)T + (-1.68 + 28.9i)T^{2} \) |
| 31 | \( 1 + (-0.0441 + 0.757i)T + (-30.7 - 3.59i)T^{2} \) |
| 37 | \( 1 + (-7.99 + 2.91i)T + (28.3 - 23.7i)T^{2} \) |
| 41 | \( 1 + (3.25 + 0.771i)T + (36.6 + 18.4i)T^{2} \) |
| 43 | \( 1 + (-0.269 - 0.625i)T + (-29.5 + 31.2i)T^{2} \) |
| 47 | \( 1 + (-0.170 - 2.93i)T + (-46.6 + 5.45i)T^{2} \) |
| 53 | \( 1 + (-2.13 + 3.69i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-6.02 + 0.703i)T + (57.4 - 13.6i)T^{2} \) |
| 61 | \( 1 + (6.40 + 3.21i)T + (36.4 + 48.9i)T^{2} \) |
| 67 | \( 1 + (2.14 - 2.27i)T + (-3.89 - 66.8i)T^{2} \) |
| 71 | \( 1 + (-2.02 - 1.69i)T + (12.3 + 69.9i)T^{2} \) |
| 73 | \( 1 + (-4.57 + 3.83i)T + (12.6 - 71.8i)T^{2} \) |
| 79 | \( 1 + (-0.977 + 0.231i)T + (70.5 - 35.4i)T^{2} \) |
| 83 | \( 1 + (8.83 - 2.09i)T + (74.1 - 37.2i)T^{2} \) |
| 89 | \( 1 + (7.18 - 6.02i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (8.36 - 11.2i)T + (-27.8 - 92.9i)T^{2} \) |
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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
−15.26807196509948246626285930310, −14.27583554684652249394128732463, −12.13787029793548706614852540786, −10.92361658735441525489809697783, −9.927792281322630074451652204082, −9.409488662756448197133048114608, −8.392381566213836739370528745444, −6.75086631992391153284277902261, −5.77259518420933578713705021144, −2.86623895812375183736753856668,
0.862365862380423436478625337151, 2.84432168547568377824903456870, 6.35204129401147415661417192576, 7.23333654551584002925284002097, 8.401434551165648282695425825844, 9.414826331228262702316419302265, 10.42731413379351548592749248593, 11.54750573496166976887737745572, 12.74316879907566738982177985293, 13.45332428194377704235461472233
