L(s) = 1 | + (−0.173 + 0.984i)2-s + (0.766 + 0.642i)3-s + (−0.939 − 0.342i)4-s + (−0.766 + 0.642i)6-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)8-s + (−0.347 − 1.96i)9-s + (3 − 5.19i)11-s + (−0.499 − 0.866i)12-s + (3.83 − 3.21i)13-s + (−0.939 + 0.342i)14-s + (0.766 + 0.642i)16-s + (0.520 − 2.95i)17-s + 2·18-s + (−0.173 + 0.984i)21-s + (4.59 + 3.85i)22-s + ⋯ |
L(s) = 1 | + (−0.122 + 0.696i)2-s + (0.442 + 0.371i)3-s + (−0.469 − 0.171i)4-s + (−0.312 + 0.262i)6-s + (0.188 + 0.327i)7-s + (0.176 − 0.306i)8-s + (−0.115 − 0.656i)9-s + (0.904 − 1.56i)11-s + (−0.144 − 0.249i)12-s + (1.06 − 0.891i)13-s + (−0.251 + 0.0914i)14-s + (0.191 + 0.160i)16-s + (0.126 − 0.716i)17-s + 0.471·18-s + (−0.0378 + 0.214i)21-s + (0.979 + 0.822i)22-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 722 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.949 - 0.313i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.65161 + 0.265486i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.65161 + 0.265486i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 + (0.173 - 0.984i)T \) |
| 19 | \( 1 \) |
good | 3 | \( 1 + (-0.766 - 0.642i)T + (0.520 + 2.95i)T^{2} \) |
| 5 | \( 1 + (3.83 - 3.21i)T^{2} \) |
| 7 | \( 1 + (-0.5 - 0.866i)T + (-3.5 + 6.06i)T^{2} \) |
| 11 | \( 1 + (-3 + 5.19i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.83 + 3.21i)T + (2.25 - 12.8i)T^{2} \) |
| 17 | \( 1 + (-0.520 + 2.95i)T + (-15.9 - 5.81i)T^{2} \) |
| 23 | \( 1 + (2.81 + 1.02i)T + (17.6 + 14.7i)T^{2} \) |
| 29 | \( 1 + (-1.56 - 8.86i)T + (-27.2 + 9.91i)T^{2} \) |
| 31 | \( 1 + (-2 - 3.46i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 2T + 37T^{2} \) |
| 41 | \( 1 + (7.11 + 40.3i)T^{2} \) |
| 43 | \( 1 + (7.51 - 2.73i)T + (32.9 - 27.6i)T^{2} \) |
| 47 | \( 1 + (-44.1 + 16.0i)T^{2} \) |
| 53 | \( 1 + (-2.81 - 1.02i)T + (40.6 + 34.0i)T^{2} \) |
| 59 | \( 1 + (-1.56 + 8.86i)T + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (-9.39 - 3.42i)T + (46.7 + 39.2i)T^{2} \) |
| 67 | \( 1 + (-0.868 - 4.92i)T + (-62.9 + 22.9i)T^{2} \) |
| 71 | \( 1 + (-5.63 + 2.05i)T + (54.3 - 45.6i)T^{2} \) |
| 73 | \( 1 + (5.36 + 4.49i)T + (12.6 + 71.8i)T^{2} \) |
| 79 | \( 1 + (7.66 + 6.42i)T + (13.7 + 77.7i)T^{2} \) |
| 83 | \( 1 + (-3 - 5.19i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (9.19 - 7.71i)T + (15.4 - 87.6i)T^{2} \) |
| 97 | \( 1 + (1.73 - 9.84i)T + (-91.1 - 33.1i)T^{2} \) |
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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
−10.27029296520396596097533441447, −9.296232797736166783810351387223, −8.614259664507290877585102324232, −8.207976309412887826182227420666, −6.81781867058067644682453051994, −6.04476167328159055900701370646, −5.26048106056144150817686825500, −3.77765938451211476147794712054, −3.19386627895935498053782102512, −0.994987878445128122932214103324,
1.55818474055833538056788055642, 2.27413665242578004865924568394, 3.96171804786003108649136739563, 4.41375252937378254555112539149, 5.97964615773022321065727361789, 7.01802220954580161291552755924, 7.958551488641049551479082112669, 8.618335687856143581723337852666, 9.693731900426290115912599349462, 10.21067785969645760652692510027