| L(s) = 1 | + (−1.98 − 1.06i)3-s + (−0.212 + 0.259i)5-s + (0.792 + 0.529i)7-s + (1.14 + 1.71i)9-s + (−2.34 − 0.709i)11-s + (−2.81 + 2.30i)13-s + (0.696 − 0.288i)15-s + (−1.82 − 0.756i)17-s + (−0.157 + 1.60i)19-s + (−1.01 − 1.89i)21-s + (7.27 + 1.44i)23-s + (0.953 + 4.79i)25-s + (0.205 + 2.08i)27-s + (1.10 + 3.64i)29-s + (−7.16 + 7.16i)31-s + ⋯ |
| L(s) = 1 | + (−1.14 − 0.612i)3-s + (−0.0950 + 0.115i)5-s + (0.299 + 0.200i)7-s + (0.382 + 0.573i)9-s + (−0.705 − 0.214i)11-s + (−0.779 + 0.640i)13-s + (0.179 − 0.0745i)15-s + (−0.442 − 0.183i)17-s + (−0.0362 + 0.367i)19-s + (−0.220 − 0.412i)21-s + (1.51 + 0.301i)23-s + (0.190 + 0.958i)25-s + (0.0395 + 0.402i)27-s + (0.205 + 0.677i)29-s + (−1.28 + 1.28i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 512 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.102 - 0.994i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.309952 + 0.343508i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.309952 + 0.343508i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| good | 3 | \( 1 + (1.98 + 1.06i)T + (1.66 + 2.49i)T^{2} \) |
| 5 | \( 1 + (0.212 - 0.259i)T + (-0.975 - 4.90i)T^{2} \) |
| 7 | \( 1 + (-0.792 - 0.529i)T + (2.67 + 6.46i)T^{2} \) |
| 11 | \( 1 + (2.34 + 0.709i)T + (9.14 + 6.11i)T^{2} \) |
| 13 | \( 1 + (2.81 - 2.30i)T + (2.53 - 12.7i)T^{2} \) |
| 17 | \( 1 + (1.82 + 0.756i)T + (12.0 + 12.0i)T^{2} \) |
| 19 | \( 1 + (0.157 - 1.60i)T + (-18.6 - 3.70i)T^{2} \) |
| 23 | \( 1 + (-7.27 - 1.44i)T + (21.2 + 8.80i)T^{2} \) |
| 29 | \( 1 + (-1.10 - 3.64i)T + (-24.1 + 16.1i)T^{2} \) |
| 31 | \( 1 + (7.16 - 7.16i)T - 31iT^{2} \) |
| 37 | \( 1 + (-0.968 + 0.0953i)T + (36.2 - 7.21i)T^{2} \) |
| 41 | \( 1 + (2.34 - 11.8i)T + (-37.8 - 15.6i)T^{2} \) |
| 43 | \( 1 + (7.65 - 4.09i)T + (23.8 - 35.7i)T^{2} \) |
| 47 | \( 1 + (-1.74 + 4.20i)T + (-33.2 - 33.2i)T^{2} \) |
| 53 | \( 1 + (-2.50 + 8.26i)T + (-44.0 - 29.4i)T^{2} \) |
| 59 | \( 1 + (-2.07 - 1.69i)T + (11.5 + 57.8i)T^{2} \) |
| 61 | \( 1 + (-3.63 + 6.80i)T + (-33.8 - 50.7i)T^{2} \) |
| 67 | \( 1 + (5.45 - 10.1i)T + (-37.2 - 55.7i)T^{2} \) |
| 71 | \( 1 + (1.79 - 2.68i)T + (-27.1 - 65.5i)T^{2} \) |
| 73 | \( 1 + (-2.04 + 1.36i)T + (27.9 - 67.4i)T^{2} \) |
| 79 | \( 1 + (4.58 + 11.0i)T + (-55.8 + 55.8i)T^{2} \) |
| 83 | \( 1 + (-7.11 - 0.700i)T + (81.4 + 16.1i)T^{2} \) |
| 89 | \( 1 + (13.6 - 2.71i)T + (82.2 - 34.0i)T^{2} \) |
| 97 | \( 1 + (10.5 - 10.5i)T - 97iT^{2} \) |
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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
−11.32019937304405684286876514243, −10.54125093152448293232934047437, −9.390280732949855499754891490249, −8.385048058952223894833464975332, −7.13799541216675556255597414902, −6.75281960189256666106200556623, −5.37775330957237541984411084699, −4.95434296881475992982815570965, −3.16235270410965993618134599635, −1.54364328418627638091413158338,
0.31718027498446749031786963388, 2.54519580265634229742700865662, 4.23526414472579723232723704044, 5.01705780339648525121748946598, 5.73884363290779457773289045572, 6.95121967828083707271109028864, 7.87492776855280398579391428152, 9.011942935571932740395959961889, 10.09742006344888226603050995910, 10.70045494033157291212894566974
