L(s) = 1 | + (0.927 − 1.06i)2-s + (−0.280 − 1.98i)4-s + (2.18 − 0.468i)5-s + 3.02·7-s + (−2.37 − 1.53i)8-s + (1.52 − 2.76i)10-s + 3.62i·11-s − 1.69·13-s + (2.80 − 3.22i)14-s + (−3.84 + 1.11i)16-s − 6.60·17-s + 5.12·19-s + (−1.54 − 4.19i)20-s + (3.86 + 3.35i)22-s − 6.67i·23-s + ⋯ |
L(s) = 1 | + (0.655 − 0.755i)2-s + (−0.140 − 0.990i)4-s + (0.977 − 0.209i)5-s + 1.14·7-s + (−0.839 − 0.543i)8-s + (0.482 − 0.875i)10-s + 1.09i·11-s − 0.470·13-s + (0.748 − 0.862i)14-s + (−0.960 + 0.277i)16-s − 1.60·17-s + 1.17·19-s + (−0.344 − 0.938i)20-s + (0.824 + 0.716i)22-s − 1.39i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 360 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.249 + 0.968i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.73670 - 1.34575i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.73670 - 1.34575i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (-0.927 + 1.06i)T \) |
| 3 | \( 1 \) |
| 5 | \( 1 + (-2.18 + 0.468i)T \) |
good | 7 | \( 1 - 3.02T + 7T^{2} \) |
| 11 | \( 1 - 3.62iT - 11T^{2} \) |
| 13 | \( 1 + 1.69T + 13T^{2} \) |
| 17 | \( 1 + 6.60T + 17T^{2} \) |
| 19 | \( 1 - 5.12T + 19T^{2} \) |
| 23 | \( 1 + 6.67iT - 23T^{2} \) |
| 29 | \( 1 + 6.82T + 29T^{2} \) |
| 31 | \( 1 - 1.73iT - 31T^{2} \) |
| 37 | \( 1 + 0.371T + 37T^{2} \) |
| 41 | \( 1 - 5.83iT - 41T^{2} \) |
| 43 | \( 1 - 5.24iT - 43T^{2} \) |
| 47 | \( 1 + 0.525iT - 47T^{2} \) |
| 53 | \( 1 - 10.0iT - 53T^{2} \) |
| 59 | \( 1 - 4.86iT - 59T^{2} \) |
| 61 | \( 1 - 61T^{2} \) |
| 67 | \( 1 - 13.4iT - 67T^{2} \) |
| 71 | \( 1 - 2.45T + 71T^{2} \) |
| 73 | \( 1 + 14.5iT - 73T^{2} \) |
| 79 | \( 1 + 14.1iT - 79T^{2} \) |
| 83 | \( 1 - 5.79T + 83T^{2} \) |
| 89 | \( 1 + 10.2iT - 89T^{2} \) |
| 97 | \( 1 - 9.33iT - 97T^{2} \) |
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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
−11.28018768877054263606671756809, −10.47513516405170671963685981375, −9.582949529990553051025455515481, −8.810712012441947776661365029202, −7.30711110780857846551093464107, −6.16310825960318586195727224147, −4.95212896810919405113163276411, −4.50486785359579443612138864569, −2.54894514897133194855472053044, −1.62515988277951906713058806243,
2.13624341081602281442419021355, 3.61607289766875547149178244059, 5.08301756843702046475124345359, 5.61364050124735744638003224449, 6.78517604147374584039744828199, 7.71648949845135433309287528638, 8.738411610346438107959119657053, 9.549116718444608133342572657960, 11.11584101070880319080755532665, 11.46876624449463907848715103510