L(s) = 1 | + (−0.117 + 1.40i)2-s + (1.66 + 0.488i)3-s + (−1.97 − 0.331i)4-s + (−1.82 + 1.53i)5-s + (−0.883 + 2.28i)6-s + (0.455 + 1.25i)7-s + (0.698 − 2.74i)8-s + (2.52 + 1.62i)9-s + (−1.94 − 2.75i)10-s + (−3.07 + 3.66i)11-s + (−3.11 − 1.51i)12-s + (0.552 − 0.0973i)13-s + (−1.81 + 0.494i)14-s + (−3.79 + 1.65i)15-s + (3.78 + 1.30i)16-s + (0.710 − 0.410i)17-s + ⋯ |
L(s) = 1 | + (−0.0830 + 0.996i)2-s + (0.959 + 0.281i)3-s + (−0.986 − 0.165i)4-s + (−0.818 + 0.686i)5-s + (−0.360 + 0.932i)6-s + (0.171 + 0.472i)7-s + (0.246 − 0.969i)8-s + (0.841 + 0.540i)9-s + (−0.616 − 0.872i)10-s + (−0.928 + 1.10i)11-s + (−0.899 − 0.436i)12-s + (0.153 − 0.0270i)13-s + (−0.485 + 0.132i)14-s + (−0.978 + 0.428i)15-s + (0.945 + 0.326i)16-s + (0.172 − 0.0995i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 216 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.685 - 0.728i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.490398 + 1.13450i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.490398 + 1.13450i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.117 - 1.40i)T \) |
| 3 | \( 1 + (-1.66 - 0.488i)T \) |
good | 5 | \( 1 + (1.82 - 1.53i)T + (0.868 - 4.92i)T^{2} \) |
| 7 | \( 1 + (-0.455 - 1.25i)T + (-5.36 + 4.49i)T^{2} \) |
| 11 | \( 1 + (3.07 - 3.66i)T + (-1.91 - 10.8i)T^{2} \) |
| 13 | \( 1 + (-0.552 + 0.0973i)T + (12.2 - 4.44i)T^{2} \) |
| 17 | \( 1 + (-0.710 + 0.410i)T + (8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-3.02 + 5.23i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (-4.41 - 1.60i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-0.586 + 3.32i)T + (-27.2 - 9.91i)T^{2} \) |
| 31 | \( 1 + (-3.71 + 10.2i)T + (-23.7 - 19.9i)T^{2} \) |
| 37 | \( 1 + (7.25 - 4.18i)T + (18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (4.16 - 0.734i)T + (38.5 - 14.0i)T^{2} \) |
| 43 | \( 1 + (-2.38 - 2.00i)T + (7.46 + 42.3i)T^{2} \) |
| 47 | \( 1 + (-10.7 + 3.89i)T + (36.0 - 30.2i)T^{2} \) |
| 53 | \( 1 + 0.541T + 53T^{2} \) |
| 59 | \( 1 + (-4.04 - 4.81i)T + (-10.2 + 58.1i)T^{2} \) |
| 61 | \( 1 + (-3.02 - 8.29i)T + (-46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (0.850 + 4.82i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (2.20 + 3.81i)T + (-35.5 + 61.4i)T^{2} \) |
| 73 | \( 1 + (0.205 - 0.355i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (4.56 + 0.804i)T + (74.2 + 27.0i)T^{2} \) |
| 83 | \( 1 + (9.31 + 1.64i)T + (77.9 + 28.3i)T^{2} \) |
| 89 | \( 1 + (8.67 + 5.00i)T + (44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (6.77 + 5.68i)T + (16.8 + 95.5i)T^{2} \) |
show more | |
show less | |
\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.09067308343108267397005562131, −11.75498272454994415099310488286, −10.41299545110409972657453926328, −9.526087492497552914079739931631, −8.535278652337194823250058794778, −7.53438354919096231926139087894, −7.09235166823123813261954785473, −5.27878179623825613989118678662, −4.20450655708297853978851124445, −2.79423807520976267300613257909,
1.11100662975302285931078655437, 3.03308631372339964239736285438, 3.94252535061514060460647261448, 5.24828454354774080600021234999, 7.37447292052968237335810676913, 8.375958899632988750729777638270, 8.731244034098384589943593767357, 10.14592412271467344913681996837, 10.93156316569794862980037577660, 12.23084712565427380688498026858