L(s) = 1 | + (−0.636 + 0.367i)2-s + (1.02 + 0.592i)3-s + (−0.730 + 1.26i)4-s + (0.136 − 0.0789i)5-s − 0.871·6-s + (1.12 + 2.39i)7-s − 2.54i·8-s + (−0.796 − 1.38i)9-s + (−0.0579 + 0.100i)10-s + (3.30 + 0.224i)11-s + (−1.5 + 0.866i)12-s − 2.14·13-s + (−1.59 − 1.10i)14-s + 0.187·15-s + (−0.527 − 0.912i)16-s + (2.20 − 3.81i)17-s + ⋯ |
L(s) = 1 | + (−0.449 + 0.259i)2-s + (0.592 + 0.342i)3-s + (−0.365 + 0.632i)4-s + (0.0611 − 0.0352i)5-s − 0.355·6-s + (0.426 + 0.904i)7-s − 0.898i·8-s + (−0.265 − 0.460i)9-s + (−0.0183 + 0.0317i)10-s + (0.997 + 0.0675i)11-s + (−0.433 + 0.249i)12-s − 0.594·13-s + (−0.426 − 0.295i)14-s + 0.0483·15-s + (−0.131 − 0.228i)16-s + (0.533 − 0.924i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 77 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.520 - 0.854i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.745017 + 0.418621i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.745017 + 0.418621i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-1.12 - 2.39i)T \) |
| 11 | \( 1 + (-3.30 - 0.224i)T \) |
good | 2 | \( 1 + (0.636 - 0.367i)T + (1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-1.02 - 0.592i)T + (1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.136 + 0.0789i)T + (2.5 - 4.33i)T^{2} \) |
| 13 | \( 1 + 2.14T + 13T^{2} \) |
| 17 | \( 1 + (-2.20 + 3.81i)T + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (3.07 + 5.32i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (1.20 + 2.08i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 - 2.10iT - 29T^{2} \) |
| 31 | \( 1 + (-3.69 - 2.13i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (-3.46 - 5.99i)T + (-18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + 6.42T + 41T^{2} \) |
| 43 | \( 1 - 6.98iT - 43T^{2} \) |
| 47 | \( 1 + (-3.21 + 1.85i)T + (23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-0.160 + 0.277i)T + (-26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 + (11.6 + 6.74i)T + (29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (3.41 + 5.91i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.32 + 2.29i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + 3.51T + 71T^{2} \) |
| 73 | \( 1 + (1.76 - 3.05i)T + (-36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + (-1.30 + 0.754i)T + (39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 - 9.49T + 83T^{2} \) |
| 89 | \( 1 + (11.7 - 6.78i)T + (44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 3.55iT - 97T^{2} \) |
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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
−14.79793600014674967663752526353, −13.76703230770452832240086041680, −12.37665928855377534231176068200, −11.59638308363307733063816399181, −9.594711123936414663120723075192, −9.061962188682110138968765099633, −8.087275340193326827635604172251, −6.63039694468354113446666479777, −4.67615999067173941140890173933, −3.03963736264974404112499150937,
1.77699599686626116964957436387, 4.18644697230226237664688137513, 5.94221540840041894495816590548, 7.69157146491475650574728671165, 8.581888769388698918567736427064, 9.924897517013777594326292846870, 10.72190260087231633057516590635, 12.05076670688752084545999915381, 13.59313121571371349146402657082, 14.25262982377426169437656699912