| L(s) = 1 | + (−1.22 + 0.707i)2-s + (−0.866 + 1.5i)3-s + (0.999 − 1.73i)4-s − 2.44i·6-s + (−6.74 − 1.88i)7-s + 2.82i·8-s + (−1.5 − 2.59i)9-s + (1.65 − 2.87i)11-s + (1.73 + 2.99i)12-s − 19.5·13-s + (9.58 − 2.45i)14-s + (−2.00 − 3.46i)16-s + (4.21 − 7.30i)17-s + (3.67 + 2.12i)18-s + (−0.704 + 0.406i)19-s + ⋯ |
| L(s) = 1 | + (−0.612 + 0.353i)2-s + (−0.288 + 0.5i)3-s + (0.249 − 0.433i)4-s − 0.408i·6-s + (−0.963 − 0.269i)7-s + 0.353i·8-s + (−0.166 − 0.288i)9-s + (0.150 − 0.260i)11-s + (0.144 + 0.249i)12-s − 1.50·13-s + (0.684 − 0.175i)14-s + (−0.125 − 0.216i)16-s + (0.247 − 0.429i)17-s + (0.204 + 0.117i)18-s + (−0.0370 + 0.0214i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1050 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & (0.572 - 0.819i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.7358791789\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.7358791789\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 + (1.22 - 0.707i)T \) |
| 3 | \( 1 + (0.866 - 1.5i)T \) |
| 5 | \( 1 \) |
| 7 | \( 1 + (6.74 + 1.88i)T \) |
| good | 11 | \( 1 + (-1.65 + 2.87i)T + (-60.5 - 104. i)T^{2} \) |
| 13 | \( 1 + 19.5T + 169T^{2} \) |
| 17 | \( 1 + (-4.21 + 7.30i)T + (-144.5 - 250. i)T^{2} \) |
| 19 | \( 1 + (0.704 - 0.406i)T + (180.5 - 312. i)T^{2} \) |
| 23 | \( 1 + (10.3 - 5.98i)T + (264.5 - 458. i)T^{2} \) |
| 29 | \( 1 + 4.68T + 841T^{2} \) |
| 31 | \( 1 + (-27.9 - 16.1i)T + (480.5 + 832. i)T^{2} \) |
| 37 | \( 1 + (5.21 - 3.01i)T + (684.5 - 1.18e3i)T^{2} \) |
| 41 | \( 1 + 19.6iT - 1.68e3T^{2} \) |
| 43 | \( 1 + 2.53iT - 1.84e3T^{2} \) |
| 47 | \( 1 + (-9.14 - 15.8i)T + (-1.10e3 + 1.91e3i)T^{2} \) |
| 53 | \( 1 + (-27.6 - 15.9i)T + (1.40e3 + 2.43e3i)T^{2} \) |
| 59 | \( 1 + (64.3 + 37.1i)T + (1.74e3 + 3.01e3i)T^{2} \) |
| 61 | \( 1 + (-95.1 + 54.9i)T + (1.86e3 - 3.22e3i)T^{2} \) |
| 67 | \( 1 + (-81.9 - 47.3i)T + (2.24e3 + 3.88e3i)T^{2} \) |
| 71 | \( 1 - 98.0T + 5.04e3T^{2} \) |
| 73 | \( 1 + (-5.55 + 9.62i)T + (-2.66e3 - 4.61e3i)T^{2} \) |
| 79 | \( 1 + (-0.500 - 0.867i)T + (-3.12e3 + 5.40e3i)T^{2} \) |
| 83 | \( 1 + 66.8T + 6.88e3T^{2} \) |
| 89 | \( 1 + (133. - 77.0i)T + (3.96e3 - 6.85e3i)T^{2} \) |
| 97 | \( 1 - 132.T + 9.40e3T^{2} \) |
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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
−9.813783842437666195187420156020, −9.262746937327376994055291952154, −8.222969756786438334490230147794, −7.25668120481479528574843299573, −6.61002816643904647123322801358, −5.64422800341817292922775173662, −4.77201562060333535720884001764, −3.57575428218490603099478940597, −2.45125715524793450881036518256, −0.63288637673654105494594481147,
0.49963430180557773022503691454, 2.05102010324093861869110707067, 2.90441219666775369512535962210, 4.18599623273266509084057495655, 5.42649384344327120816269707913, 6.42356640647578383724833091083, 7.11476309296472396062850498583, 7.928267621835333638984272912402, 8.833751939963192822848538983745, 9.877797686117742477569438468700
