| L(s) = 1 | + (−29.0 − 13.4i)2-s − 161. i·3-s + (660. + 782. i)4-s + (462. + 3.09e3i)5-s + (−2.17e3 + 4.69e3i)6-s + (−1.32e4 + 1.32e4i)7-s + (−8.63e3 − 3.16e4i)8-s + 3.29e4·9-s + (2.82e4 − 9.59e4i)10-s + (1.45e5 + 1.45e5i)11-s + (1.26e5 − 1.06e5i)12-s − 9.30e4i·13-s + (5.64e5 − 2.06e5i)14-s + (4.99e5 − 7.46e4i)15-s + (−1.75e5 + 1.03e6i)16-s + (4.81e5 + 4.81e5i)17-s + ⋯ |
| L(s) = 1 | + (−0.906 − 0.421i)2-s − 0.665i·3-s + (0.645 + 0.763i)4-s + (0.147 + 0.989i)5-s + (−0.280 + 0.603i)6-s + (−0.790 + 0.790i)7-s + (−0.263 − 0.964i)8-s + 0.557·9-s + (0.282 − 0.959i)10-s + (0.906 + 0.906i)11-s + (0.508 − 0.429i)12-s − 0.250i·13-s + (1.04 − 0.384i)14-s + (0.657 − 0.0983i)15-s + (−0.167 + 0.985i)16-s + (0.338 + 0.338i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(11-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 80 ^{s/2} \, \Gamma_{\C}(s+5) \, L(s)\cr =\mathstrut & (-0.140 - 0.990i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{11}{2})\) |
\(\approx\) |
\(0.671983 + 0.774261i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.671983 + 0.774261i\) |
| \(L(6)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (29.0 + 13.4i)T \) |
| 5 | \( 1 + (-462. - 3.09e3i)T \) |
| good | 3 | \( 1 + 161. iT - 5.90e4T^{2} \) |
| 7 | \( 1 + (1.32e4 - 1.32e4i)T - 2.82e8iT^{2} \) |
| 11 | \( 1 + (-1.45e5 - 1.45e5i)T + 2.59e10iT^{2} \) |
| 13 | \( 1 + 9.30e4iT - 1.37e11T^{2} \) |
| 17 | \( 1 + (-4.81e5 - 4.81e5i)T + 2.01e12iT^{2} \) |
| 19 | \( 1 + (-1.62e6 - 1.62e6i)T + 6.13e12iT^{2} \) |
| 23 | \( 1 + (4.96e5 + 4.96e5i)T + 4.14e13iT^{2} \) |
| 29 | \( 1 + (1.67e7 + 1.67e7i)T + 4.20e14iT^{2} \) |
| 31 | \( 1 - 2.35e7T + 8.19e14T^{2} \) |
| 37 | \( 1 - 8.53e7iT - 4.80e15T^{2} \) |
| 41 | \( 1 - 1.28e7iT - 1.34e16T^{2} \) |
| 43 | \( 1 + 9.53e7T + 2.16e16T^{2} \) |
| 47 | \( 1 + (1.87e8 + 1.87e8i)T + 5.25e16iT^{2} \) |
| 53 | \( 1 - 6.79e6T + 1.74e17T^{2} \) |
| 59 | \( 1 + (1.99e7 - 1.99e7i)T - 5.11e17iT^{2} \) |
| 61 | \( 1 + (-1.02e8 + 1.02e8i)T - 7.13e17iT^{2} \) |
| 67 | \( 1 + 1.06e9T + 1.82e18T^{2} \) |
| 71 | \( 1 - 2.56e9iT - 3.25e18T^{2} \) |
| 73 | \( 1 + (-2.17e9 - 2.17e9i)T + 4.29e18iT^{2} \) |
| 79 | \( 1 + 1.19e8iT - 9.46e18T^{2} \) |
| 83 | \( 1 - 4.47e9iT - 1.55e19T^{2} \) |
| 89 | \( 1 - 8.10e9T + 3.11e19T^{2} \) |
| 97 | \( 1 + (-3.22e9 - 3.22e9i)T + 7.37e19iT^{2} \) |
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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
−12.31302166465327373311264859527, −11.66164650322876659922564363713, −10.06710584427668177103184145497, −9.634643904225241191886891661687, −8.034315224967616414385437086511, −6.96171017763889921212415491596, −6.22988143044166673913093981844, −3.66143919087913836529111879765, −2.41755059621461297365051150033, −1.36372019773750637170095770342,
0.40214932772154020592385831147, 1.36063055705018973641800142883, 3.55964345819000963619115566321, 4.96995944962137973226723058231, 6.34836954736227136819175727334, 7.52457509674445268878931383577, 9.033206667350689503502095185661, 9.504850498560094241654657550625, 10.56135524973677906519248252912, 11.76165525016261368503547254690
