| 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 | \( 1 + (6.77 - 5.68i)T + (16.8 - 95.5i)T^{2} \) |
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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
−12.23084712565427380688498026858, −10.93156316569794862980037577660, −10.14592412271467344913681996837, −8.731244034098384589943593767357, −8.375958899632988750729777638270, −7.37447292052968237335810676913, −5.24828454354774080600021234999, −3.94252535061514060460647261448, −3.03308631372339964239736285438, −1.11100662975302285931078655437,
2.79423807520976267300613257909, 4.20450655708297853978851124445, 5.27878179623825613989118678662, 7.09235166823123813261954785473, 7.53438354919096231926139087894, 8.535278652337194823250058794778, 9.526087492497552914079739931631, 10.41299545110409972657453926328, 11.75498272454994415099310488286, 13.09067308343108267397005562131
