| L(s) = 1 | + (−0.550 + 1.92i)2-s + (−3.39 − 2.11i)4-s − 2.10i·5-s + 2.64i·7-s + (5.93 − 5.36i)8-s + (4.05 + 1.15i)10-s − 5.49·11-s + 0.893i·13-s + (−5.08 − 1.45i)14-s + (7.04 + 14.3i)16-s − 10.0·17-s + 25.3·19-s + (−4.45 + 7.15i)20-s + (3.02 − 10.5i)22-s + 15.9i·23-s + ⋯ |
| L(s) = 1 | + (−0.275 + 0.961i)2-s + (−0.848 − 0.529i)4-s − 0.421i·5-s + 0.377i·7-s + (0.742 − 0.670i)8-s + (0.405 + 0.115i)10-s − 0.499·11-s + 0.0687i·13-s + (−0.363 − 0.103i)14-s + (0.440 + 0.897i)16-s − 0.590·17-s + 1.33·19-s + (−0.222 + 0.357i)20-s + (0.137 − 0.479i)22-s + 0.692i·23-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 504 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (-0.670 - 0.742i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(1.040075749\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.040075749\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (0.550 - 1.92i)T \) |
| 3 | \( 1 \) |
| 7 | \( 1 - 2.64iT \) |
| good | 5 | \( 1 + 2.10iT - 25T^{2} \) |
| 11 | \( 1 + 5.49T + 121T^{2} \) |
| 13 | \( 1 - 0.893iT - 169T^{2} \) |
| 17 | \( 1 + 10.0T + 289T^{2} \) |
| 19 | \( 1 - 25.3T + 361T^{2} \) |
| 23 | \( 1 - 15.9iT - 529T^{2} \) |
| 29 | \( 1 - 28.8iT - 841T^{2} \) |
| 31 | \( 1 - 41.5iT - 961T^{2} \) |
| 37 | \( 1 - 37.1iT - 1.36e3T^{2} \) |
| 41 | \( 1 + 70.6T + 1.68e3T^{2} \) |
| 43 | \( 1 - 6.60T + 1.84e3T^{2} \) |
| 47 | \( 1 - 44.9iT - 2.20e3T^{2} \) |
| 53 | \( 1 - 27.8iT - 2.80e3T^{2} \) |
| 59 | \( 1 - 38.7T + 3.48e3T^{2} \) |
| 61 | \( 1 + 7.21iT - 3.72e3T^{2} \) |
| 67 | \( 1 + 10.9T + 4.48e3T^{2} \) |
| 71 | \( 1 + 72.4iT - 5.04e3T^{2} \) |
| 73 | \( 1 - 26.0T + 5.32e3T^{2} \) |
| 79 | \( 1 - 96.8iT - 6.24e3T^{2} \) |
| 83 | \( 1 + 26.3T + 6.88e3T^{2} \) |
| 89 | \( 1 + 41.8T + 7.92e3T^{2} \) |
| 97 | \( 1 - 74.6T + 9.40e3T^{2} \) |
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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
−10.83962536602708054900480115989, −9.904598344275966132576662801341, −9.038454252377643388733441241647, −8.397355098534684306545544171087, −7.39134836946024026960159231344, −6.55856208015327534177525541296, −5.33675861691110928478949413957, −4.84442136726657911040293367818, −3.26451096085391709840264637521, −1.30398301322141548102808581819,
0.50830855753307443950393288064, 2.18782871702009216214496878228, 3.26291332308856727933292738867, 4.36576298036437668881090902511, 5.46890051031178085196395861943, 6.92485290522576399769526620651, 7.82097976952112641127958196025, 8.737202322161111763594983184880, 9.772832258964503254840783539612, 10.37665135634493584736402894391
