L(s) = 1 | + (−0.786 − 0.618i)3-s + (−0.841 − 0.540i)4-s + (1.68 + 0.408i)7-s + (0.235 + 0.971i)9-s + (0.327 + 0.945i)12-s + (−0.173 − 0.336i)13-s + (0.415 + 0.909i)16-s + (0.839 + 1.17i)19-s + (−1.07 − 1.36i)21-s + (0.888 − 0.458i)25-s + (0.415 − 0.909i)27-s + (−1.19 − 1.25i)28-s + (−0.279 − 1.94i)31-s + (0.327 − 0.945i)36-s + (−0.771 − 0.308i)37-s + ⋯ |
L(s) = 1 | + (−0.786 − 0.618i)3-s + (−0.841 − 0.540i)4-s + (1.68 + 0.408i)7-s + (0.235 + 0.971i)9-s + (0.327 + 0.945i)12-s + (−0.173 − 0.336i)13-s + (0.415 + 0.909i)16-s + (0.839 + 1.17i)19-s + (−1.07 − 1.36i)21-s + (0.888 − 0.458i)25-s + (0.415 − 0.909i)27-s + (−1.19 − 1.25i)28-s + (−0.279 − 1.94i)31-s + (0.327 − 0.945i)36-s + (−0.771 − 0.308i)37-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1191 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.731 + 0.681i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(0.8230784568\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.8230784568\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (0.786 + 0.618i)T \) |
| 397 | \( 1 - T \) |
good | 2 | \( 1 + (0.841 + 0.540i)T^{2} \) |
| 5 | \( 1 + (-0.888 + 0.458i)T^{2} \) |
| 7 | \( 1 + (-1.68 - 0.408i)T + (0.888 + 0.458i)T^{2} \) |
| 11 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 13 | \( 1 + (0.173 + 0.336i)T + (-0.580 + 0.814i)T^{2} \) |
| 17 | \( 1 + (0.841 - 0.540i)T^{2} \) |
| 19 | \( 1 + (-0.839 - 1.17i)T + (-0.327 + 0.945i)T^{2} \) |
| 23 | \( 1 + (0.786 - 0.618i)T^{2} \) |
| 29 | \( 1 + (-0.0475 + 0.998i)T^{2} \) |
| 31 | \( 1 + (0.279 + 1.94i)T + (-0.959 + 0.281i)T^{2} \) |
| 37 | \( 1 + (0.771 + 0.308i)T + (0.723 + 0.690i)T^{2} \) |
| 41 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (-0.452 - 0.132i)T + (0.841 + 0.540i)T^{2} \) |
| 47 | \( 1 + (0.995 + 0.0950i)T^{2} \) |
| 53 | \( 1 + (-0.654 + 0.755i)T^{2} \) |
| 59 | \( 1 + (0.235 - 0.971i)T^{2} \) |
| 61 | \( 1 + (-0.566 + 1.09i)T + (-0.580 - 0.814i)T^{2} \) |
| 67 | \( 1 + (1.56 + 0.625i)T + (0.723 + 0.690i)T^{2} \) |
| 71 | \( 1 + (-0.959 + 0.281i)T^{2} \) |
| 73 | \( 1 + (-1.76 - 0.168i)T + (0.981 + 0.189i)T^{2} \) |
| 79 | \( 1 + (0.786 - 1.36i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 + (-0.841 - 0.540i)T^{2} \) |
| 89 | \( 1 + (0.580 + 0.814i)T^{2} \) |
| 97 | \( 1 + (-1.44 - 1.37i)T + (0.0475 + 0.998i)T^{2} \) |
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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
−9.985329413239007104060828732593, −8.967862900040828213470777393724, −8.019721578978938047474934766389, −7.67604212968501856279740134546, −6.29518778867131511206697788778, −5.43174955277991233009264550570, −5.04668868393526370489186732338, −4.05856663886731911673310717736, −2.14098193585063054783803207997, −1.11735714751445065116988286945,
1.22231599104511510944417701294, 3.16924641270383790485768948064, 4.27872237065334151917702716050, 4.93045154966282235430526786196, 5.33780416119972710983468329913, 6.92311319861544905257332482280, 7.52918252555183114770584729184, 8.709346455850391738837445462820, 9.031447262230448969524474706971, 10.18105204104591609341431930658