L(s) = 1 | + (0.906 + 1.08i)2-s + (0.406 + 0.703i)3-s + (−0.356 + 1.96i)4-s + (0.866 + 0.5i)5-s + (−0.395 + 1.07i)6-s + (−0.336 − 2.62i)7-s + (−2.45 + 1.39i)8-s + (1.17 − 2.02i)9-s + (0.242 + 1.39i)10-s + (−4.20 + 2.43i)11-s + (−1.52 + 0.548i)12-s + 0.895i·13-s + (2.54 − 2.74i)14-s + 0.812i·15-s + (−3.74 − 1.40i)16-s + (5.10 − 2.94i)17-s + ⋯ |
L(s) = 1 | + (0.641 + 0.767i)2-s + (0.234 + 0.406i)3-s + (−0.178 + 0.984i)4-s + (0.387 + 0.223i)5-s + (−0.161 + 0.440i)6-s + (−0.127 − 0.991i)7-s + (−0.869 + 0.494i)8-s + (0.390 − 0.675i)9-s + (0.0766 + 0.440i)10-s + (−1.26 + 0.732i)11-s + (−0.441 + 0.158i)12-s + 0.248i·13-s + (0.679 − 0.733i)14-s + 0.209i·15-s + (−0.936 − 0.350i)16-s + (1.23 − 0.714i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 140 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.177 - 0.984i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.19788 + 1.00125i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.19788 + 1.00125i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.906 - 1.08i)T \) |
| 5 | \( 1 + (-0.866 - 0.5i)T \) |
| 7 | \( 1 + (0.336 + 2.62i)T \) |
good | 3 | \( 1 + (-0.406 - 0.703i)T + (-1.5 + 2.59i)T^{2} \) |
| 11 | \( 1 + (4.20 - 2.43i)T + (5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 0.895iT - 13T^{2} \) |
| 17 | \( 1 + (-5.10 + 2.94i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-1.45 + 2.52i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-1.35 - 0.780i)T + (11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 9.73T + 29T^{2} \) |
| 31 | \( 1 + (2.20 + 3.82i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-0.910 + 1.57i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 10.4iT - 41T^{2} \) |
| 43 | \( 1 - 3.04iT - 43T^{2} \) |
| 47 | \( 1 + (-2.60 + 4.52i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (0.0898 + 0.155i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.68 - 9.84i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (5.33 + 3.07i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (6.75 - 3.89i)T + (33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 - 8.38iT - 71T^{2} \) |
| 73 | \( 1 + (7.60 - 4.39i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (3.14 + 1.81i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 - 2.03T + 83T^{2} \) |
| 89 | \( 1 + (-1.52 - 0.882i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + 7.83iT - 97T^{2} \) |
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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
−13.39169562051552116533045076906, −12.83926657412328410006301253392, −11.47752170568035810100428993009, −10.09032280728059173906747464456, −9.337178427095326446413237579624, −7.64944591410101652304013882101, −7.07394914072950095385109027822, −5.57373696745948254890564750778, −4.37607664493955247029876035536, −3.09694879268232393752217032005,
1.93161376050243317617147164647, 3.25530625207117812561911996150, 5.24795920792933093982421180419, 5.82211352576924868823178710253, 7.70371001389334530314247611172, 8.874506145706000203483686399047, 10.12344634226008662361897135585, 10.89926118748796084857781994168, 12.29133955455162276522254106258, 12.85032043373085595140364398509