| L(s) = 1 | + 1.41i·2-s + (2.36 − 1.84i)3-s − 2.00·4-s + (1.22 − 0.707i)5-s + (2.61 + 3.34i)6-s + (3.10 + 6.27i)7-s − 2.82i·8-s + (2.17 − 8.73i)9-s + (1.00 + 1.73i)10-s + (15.1 + 8.72i)11-s + (−4.72 + 3.69i)12-s + (8.59 − 14.8i)13-s + (−8.87 + 4.38i)14-s + (1.58 − 3.93i)15-s + 4.00·16-s + (−20.8 + 12.0i)17-s + ⋯ |
| L(s) = 1 | + 0.707i·2-s + (0.788 − 0.615i)3-s − 0.500·4-s + (0.244 − 0.141i)5-s + (0.435 + 0.557i)6-s + (0.443 + 0.896i)7-s − 0.353i·8-s + (0.241 − 0.970i)9-s + (0.100 + 0.173i)10-s + (1.37 + 0.793i)11-s + (−0.394 + 0.307i)12-s + (0.660 − 1.14i)13-s + (−0.633 + 0.313i)14-s + (0.105 − 0.262i)15-s + 0.250·16-s + (−1.22 + 0.708i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 126 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.899 - 0.437i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.80779 + 0.416279i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.80779 + 0.416279i\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - 1.41iT \) |
| 3 | \( 1 + (-2.36 + 1.84i)T \) |
| 7 | \( 1 + (-3.10 - 6.27i)T \) |
| good | 5 | \( 1 + (-1.22 + 0.707i)T + (12.5 - 21.6i)T^{2} \) |
| 11 | \( 1 + (-15.1 - 8.72i)T + (60.5 + 104. i)T^{2} \) |
| 13 | \( 1 + (-8.59 + 14.8i)T + (-84.5 - 146. i)T^{2} \) |
| 17 | \( 1 + (20.8 - 12.0i)T + (144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (-11.7 + 20.4i)T + (-180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (27.1 - 15.6i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + (13.7 - 7.94i)T + (420.5 - 728. i)T^{2} \) |
| 31 | \( 1 + 32.1T + 961T^{2} \) |
| 37 | \( 1 + (-15.7 + 27.1i)T + (-684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + (-8.67 - 5.00i)T + (840.5 + 1.45e3i)T^{2} \) |
| 43 | \( 1 + (31.6 + 54.8i)T + (-924.5 + 1.60e3i)T^{2} \) |
| 47 | \( 1 + 4.27iT - 2.20e3T^{2} \) |
| 53 | \( 1 + (45.3 - 26.1i)T + (1.40e3 - 2.43e3i)T^{2} \) |
| 59 | \( 1 + 42.1iT - 3.48e3T^{2} \) |
| 61 | \( 1 + 12.4T + 3.72e3T^{2} \) |
| 67 | \( 1 - 96.1T + 4.48e3T^{2} \) |
| 71 | \( 1 - 20.8iT - 5.04e3T^{2} \) |
| 73 | \( 1 + (-41.6 - 72.1i)T + (-2.66e3 + 4.61e3i)T^{2} \) |
| 79 | \( 1 + 45.1T + 6.24e3T^{2} \) |
| 83 | \( 1 + (-3.73 + 2.15i)T + (3.44e3 - 5.96e3i)T^{2} \) |
| 89 | \( 1 + (-123. - 71.4i)T + (3.96e3 + 6.85e3i)T^{2} \) |
| 97 | \( 1 + (17.5 + 30.3i)T + (-4.70e3 + 8.14e3i)T^{2} \) |
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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
−13.30423985369621039130425047068, −12.52563837140227144079292388980, −11.34595436614626427118904226095, −9.474154205303653611291041605238, −8.900667220953553962607009615543, −7.84421435522950740900840302006, −6.70695401993968504456701714332, −5.56790695286681668870777722677, −3.81907786366031975260324231777, −1.82457168331083138797884389799,
1.78505693148546906156090459139, 3.67837941461927612226489632340, 4.41269918100743273689869533840, 6.41466609326211054325451714003, 8.032300271744783177684855630740, 9.064938153724245469961299327023, 9.880460931521452262352066650042, 11.03469116953065777256246533538, 11.69703856616252849497243496161, 13.41798275944876583434430125120
