L(s) = 1 | + (−0.280 − 0.0752i)3-s + (2.21 + 0.332i)5-s + (−1.13 − 2.39i)7-s + (−2.52 − 1.45i)9-s + (−1.58 − 2.74i)11-s + (1.12 − 1.12i)13-s + (−0.595 − 0.259i)15-s + (0.781 − 2.91i)17-s + (1.03 − 1.79i)19-s + (0.137 + 0.756i)21-s + (−2.21 + 0.593i)23-s + (4.77 + 1.47i)25-s + (1.21 + 1.21i)27-s + 1.39i·29-s + (−0.467 + 0.269i)31-s + ⋯ |
L(s) = 1 | + (−0.162 − 0.0434i)3-s + (0.988 + 0.148i)5-s + (−0.427 − 0.904i)7-s + (−0.841 − 0.485i)9-s + (−0.477 − 0.826i)11-s + (0.312 − 0.312i)13-s + (−0.153 − 0.0670i)15-s + (0.189 − 0.706i)17-s + (0.237 − 0.411i)19-s + (0.0300 + 0.165i)21-s + (−0.461 + 0.123i)23-s + (0.955 + 0.294i)25-s + (0.233 + 0.233i)27-s + 0.259i·29-s + (−0.0839 + 0.0484i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 560 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.135 + 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.972267 - 0.848502i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.972267 - 0.848502i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (-2.21 - 0.332i)T \) |
| 7 | \( 1 + (1.13 + 2.39i)T \) |
good | 3 | \( 1 + (0.280 + 0.0752i)T + (2.59 + 1.5i)T^{2} \) |
| 11 | \( 1 + (1.58 + 2.74i)T + (-5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.12 + 1.12i)T - 13iT^{2} \) |
| 17 | \( 1 + (-0.781 + 2.91i)T + (-14.7 - 8.5i)T^{2} \) |
| 19 | \( 1 + (-1.03 + 1.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (2.21 - 0.593i)T + (19.9 - 11.5i)T^{2} \) |
| 29 | \( 1 - 1.39iT - 29T^{2} \) |
| 31 | \( 1 + (0.467 - 0.269i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (1.72 + 6.45i)T + (-32.0 + 18.5i)T^{2} \) |
| 41 | \( 1 - 1.82iT - 41T^{2} \) |
| 43 | \( 1 + (-6.50 - 6.50i)T + 43iT^{2} \) |
| 47 | \( 1 + (-12.4 + 3.32i)T + (40.7 - 23.5i)T^{2} \) |
| 53 | \( 1 + (-2.89 + 10.8i)T + (-45.8 - 26.5i)T^{2} \) |
| 59 | \( 1 + (2.55 + 4.43i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (12.2 + 7.09i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-3.08 - 0.826i)T + (58.0 + 33.5i)T^{2} \) |
| 71 | \( 1 + 7.61T + 71T^{2} \) |
| 73 | \( 1 + (-3.08 - 0.827i)T + (63.2 + 36.5i)T^{2} \) |
| 79 | \( 1 + (-14.7 - 8.52i)T + (39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (4.62 - 4.62i)T - 83iT^{2} \) |
| 89 | \( 1 + (8.86 - 15.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (-3.71 - 3.71i)T + 97iT^{2} \) |
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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
−10.68889922871688060215346405951, −9.663133873924555286812839563614, −8.991496439788842853798867863018, −7.85867316053894712069923760863, −6.79085035710981250823188493073, −5.96180452900161908052757864946, −5.20980159905446533439792535162, −3.61721517038262042278120994180, −2.66138807870953342523252068936, −0.74611647864878894825992484259,
1.91506115875347813929094914745, 2.86849954685561295466517507553, 4.53316412093208853744799100008, 5.76674756701180440000192747955, 5.96356087399823055936263417859, 7.37608800357636926699093391118, 8.538081502391295922968155116446, 9.171683667429921672549503121340, 10.14495464522643610391045287102, 10.74734681823454907470982183537