L(s) = 1 | + (−0.995 + 0.0950i)3-s + (−0.142 + 0.989i)4-s + (−0.981 − 0.189i)7-s + (0.981 − 0.189i)9-s + (0.0475 − 0.998i)12-s + (−1.65 + 0.660i)13-s + (−0.959 − 0.281i)16-s + (−0.473 − 0.451i)19-s + (0.995 + 0.0950i)21-s + (0.928 − 0.371i)25-s + (−0.959 + 0.281i)27-s + (0.327 − 0.945i)28-s + (−0.738 − 1.61i)31-s + (0.0475 + 0.998i)36-s + (−1.11 + 1.56i)37-s + ⋯ |
L(s) = 1 | + (−0.995 + 0.0950i)3-s + (−0.142 + 0.989i)4-s + (−0.981 − 0.189i)7-s + (0.981 − 0.189i)9-s + (0.0475 − 0.998i)12-s + (−1.65 + 0.660i)13-s + (−0.959 − 0.281i)16-s + (−0.473 − 0.451i)19-s + (0.995 + 0.0950i)21-s + (0.928 − 0.371i)25-s + (−0.959 + 0.281i)27-s + (0.327 − 0.945i)28-s + (−0.738 − 1.61i)31-s + (0.0475 + 0.998i)36-s + (−1.11 + 1.56i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.882 + 0.469i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.05379893345\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.05379893345\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.995 - 0.0950i)T \) |
| 397 | \( 1 - T \) |
good | 2 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 5 | \( 1 + (-0.928 + 0.371i)T^{2} \) |
| 7 | \( 1 + (0.981 + 0.189i)T + (0.928 + 0.371i)T^{2} \) |
| 11 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 13 | \( 1 + (1.65 - 0.660i)T + (0.723 - 0.690i)T^{2} \) |
| 17 | \( 1 + (0.142 + 0.989i)T^{2} \) |
| 19 | \( 1 + (0.473 + 0.451i)T + (0.0475 + 0.998i)T^{2} \) |
| 23 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 29 | \( 1 + (0.786 + 0.618i)T^{2} \) |
| 31 | \( 1 + (0.738 + 1.61i)T + (-0.654 + 0.755i)T^{2} \) |
| 37 | \( 1 + (1.11 - 1.56i)T + (-0.327 - 0.945i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (1.28 + 1.48i)T + (-0.142 + 0.989i)T^{2} \) |
| 47 | \( 1 + (-0.235 - 0.971i)T^{2} \) |
| 53 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 59 | \( 1 + (-0.981 - 0.189i)T^{2} \) |
| 61 | \( 1 + (1.84 + 0.739i)T + (0.723 + 0.690i)T^{2} \) |
| 67 | \( 1 + (0.165 - 0.231i)T + (-0.327 - 0.945i)T^{2} \) |
| 71 | \( 1 + (0.654 - 0.755i)T^{2} \) |
| 73 | \( 1 + (-0.437 - 1.80i)T + (-0.888 + 0.458i)T^{2} \) |
| 79 | \( 1 + (-0.995 - 1.72i)T + (-0.5 + 0.866i)T^{2} \) |
| 83 | \( 1 + (0.142 - 0.989i)T^{2} \) |
| 89 | \( 1 + (-0.723 - 0.690i)T^{2} \) |
| 97 | \( 1 + (0.154 + 0.445i)T + (-0.786 + 0.618i)T^{2} \) |
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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
−10.33046654788429063979428337348, −9.685208092924316868952362446770, −8.953129993341206305236271297182, −7.78886737324817682559018961623, −6.86822576004870449717279738299, −6.62570888590811615609596521571, −5.19507032504111143550463246306, −4.47580172419869243363265811392, −3.52140194035693050524052481856, −2.32082798726675602730080454022,
0.05056778604595899224270438857, 1.73950041327614017357660135756, 3.18612788855541061560695760285, 4.69740376779057225157350552459, 5.23487745363462233726269150265, 6.09240736537010390493587055169, 6.79716784019536180325420565236, 7.54459874743266388127539134252, 9.006295513108270945282995064857, 9.640661580191857388000832244266