L(s) = 1 | + (−0.207 + 0.358i)3-s + (−0.5 − 0.866i)5-s + (1.62 + 2.09i)7-s + (1.41 + 2.44i)9-s + (−0.414 + 0.717i)11-s + 2·13-s + 0.414·15-s + (3.82 − 6.63i)17-s + (2.82 + 4.89i)19-s + (−1.08 + 0.148i)21-s + (2.79 + 4.83i)23-s + (−0.499 + 0.866i)25-s − 2.41·27-s − 7.82·29-s + (−0.414 + 0.717i)31-s + ⋯ |
L(s) = 1 | + (−0.119 + 0.207i)3-s + (−0.223 − 0.387i)5-s + (0.612 + 0.790i)7-s + (0.471 + 0.816i)9-s + (−0.124 + 0.216i)11-s + 0.554·13-s + 0.106·15-s + (0.928 − 1.60i)17-s + (0.648 + 1.12i)19-s + (−0.236 + 0.0324i)21-s + (0.582 + 1.00i)23-s + (−0.0999 + 0.173i)25-s − 0.464·27-s − 1.45·29-s + (−0.0743 + 0.128i)31-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 280 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.827 - 0.561i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.25352 + 0.385170i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.25352 + 0.385170i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 5 | \( 1 + (0.5 + 0.866i)T \) |
| 7 | \( 1 + (-1.62 - 2.09i)T \) |
good | 3 | \( 1 + (0.207 - 0.358i)T + (-1.5 - 2.59i)T^{2} \) |
| 11 | \( 1 + (0.414 - 0.717i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 - 2T + 13T^{2} \) |
| 17 | \( 1 + (-3.82 + 6.63i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-2.82 - 4.89i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (-2.79 - 4.83i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + 7.82T + 29T^{2} \) |
| 31 | \( 1 + (0.414 - 0.717i)T + (-15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 + (2.82 + 4.89i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 - 5.82T + 41T^{2} \) |
| 43 | \( 1 + 6.89T + 43T^{2} \) |
| 47 | \( 1 + (5.82 + 10.0i)T + (-23.5 + 40.7i)T^{2} \) |
| 53 | \( 1 + (-2.82 + 4.89i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (-2 + 3.46i)T + (-29.5 - 51.0i)T^{2} \) |
| 61 | \( 1 + (3.32 + 5.76i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-6.44 + 11.1i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 12T + 71T^{2} \) |
| 73 | \( 1 + (1.82 - 3.16i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-2 - 3.46i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + 4.75T + 83T^{2} \) |
| 89 | \( 1 + (-2.67 - 4.62i)T + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 - 6T + 97T^{2} \) |
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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
−11.78728730566358391678772207441, −11.22509027883141030294383322688, −9.970958941302115474549782591068, −9.175301682137778499968945261450, −7.990939318172307776076728691178, −7.33317776428191700037641836602, −5.50931223384563075237847348737, −5.06629752891740358246001047760, −3.52318138277711229973587567654, −1.75438787182776435296767405066,
1.24851763638738315928648747943, 3.34266005373798626897546140278, 4.38171337426815920232364466510, 5.88126113513778346760081005637, 6.92504814515629327873159953462, 7.77056097420570372136053467131, 8.829637360490311665116719758458, 10.06917782643637430410491447252, 10.88309042528597996546691812297, 11.64029908082314514060507090430