| L(s) = 1 | + (−0.350 + 1.65i)2-s + (−1.46 − 1.62i)3-s + (−0.774 − 0.344i)4-s − 3.64i·5-s + (3.19 − 1.84i)6-s + (−1.03 + 2.32i)7-s + (−1.14 + 1.57i)8-s + (−0.184 + 1.75i)9-s + (6.01 + 1.27i)10-s + (2.31 − 0.243i)11-s + (0.571 + 1.76i)12-s + (−2.98 + 2.02i)13-s + (−3.48 − 2.52i)14-s + (−5.91 + 5.32i)15-s + (−3.33 − 3.69i)16-s + (0.442 − 4.20i)17-s + ⋯ |
| L(s) = 1 | + (−0.248 + 1.16i)2-s + (−0.843 − 0.936i)3-s + (−0.387 − 0.172i)4-s − 1.62i·5-s + (1.30 − 0.752i)6-s + (−0.392 + 0.880i)7-s + (−0.404 + 0.556i)8-s + (−0.0616 + 0.586i)9-s + (1.90 + 0.404i)10-s + (0.697 − 0.0733i)11-s + (0.165 + 0.508i)12-s + (−0.827 + 0.561i)13-s + (−0.930 − 0.675i)14-s + (−1.52 + 1.37i)15-s + (−0.832 − 0.924i)16-s + (0.107 − 1.02i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 403 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.539 + 0.842i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.147021 - 0.268720i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.147021 - 0.268720i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 13 | \( 1 + (2.98 - 2.02i)T \) |
| 31 | \( 1 + (5.10 + 2.21i)T \) |
| good | 2 | \( 1 + (0.350 - 1.65i)T + (-1.82 - 0.813i)T^{2} \) |
| 3 | \( 1 + (1.46 + 1.62i)T + (-0.313 + 2.98i)T^{2} \) |
| 5 | \( 1 + 3.64iT - 5T^{2} \) |
| 7 | \( 1 + (1.03 - 2.32i)T + (-4.68 - 5.20i)T^{2} \) |
| 11 | \( 1 + (-2.31 + 0.243i)T + (10.7 - 2.28i)T^{2} \) |
| 17 | \( 1 + (-0.442 + 4.20i)T + (-16.6 - 3.53i)T^{2} \) |
| 19 | \( 1 + (5.78 + 5.21i)T + (1.98 + 18.8i)T^{2} \) |
| 23 | \( 1 + (6.56 - 2.92i)T + (15.3 - 17.0i)T^{2} \) |
| 29 | \( 1 + (-0.840 - 0.178i)T + (26.4 + 11.7i)T^{2} \) |
| 37 | \( 1 + (-4.99 - 2.88i)T + (18.5 + 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.567 + 2.66i)T + (-37.4 - 16.6i)T^{2} \) |
| 43 | \( 1 + (-0.967 + 1.07i)T + (-4.49 - 42.7i)T^{2} \) |
| 47 | \( 1 + (1.41 + 0.460i)T + (38.0 + 27.6i)T^{2} \) |
| 53 | \( 1 + (5.51 + 4.01i)T + (16.3 + 50.4i)T^{2} \) |
| 59 | \( 1 + (-2.64 - 12.4i)T + (-53.8 + 23.9i)T^{2} \) |
| 61 | \( 1 + (6.87 + 11.9i)T + (-30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-12.7 - 7.37i)T + (33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (-2.21 - 0.232i)T + (69.4 + 14.7i)T^{2} \) |
| 73 | \( 1 + (4.38 + 6.03i)T + (-22.5 + 69.4i)T^{2} \) |
| 79 | \( 1 + (8.91 + 6.47i)T + (24.4 + 75.1i)T^{2} \) |
| 83 | \( 1 + (-10.1 + 3.30i)T + (67.1 - 48.7i)T^{2} \) |
| 89 | \( 1 + (-9.37 + 0.985i)T + (87.0 - 18.5i)T^{2} \) |
| 97 | \( 1 + (0.222 - 0.499i)T + (-64.9 - 72.0i)T^{2} \) |
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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
−11.51604214493955847028165483602, −9.358607119768761191470627968894, −9.089383321135489671685023168250, −8.020393120810057706493385118607, −7.05913092278874571793511702306, −6.23036432246005594935164170096, −5.51943536820191279881492487451, −4.59099355975482130057451518903, −2.08885114516746567014161775103, −0.22186554727300255668833241254,
2.18529407762232654394103746323, 3.61929093655555548695374578520, 4.11749428782158366524030649520, 6.04978308846071041042847351139, 6.58401079984259628785387753058, 7.87017447349255448862073460988, 9.622773120271174908078009878344, 10.29938203475069090212213707441, 10.51502920905973263430083995270, 11.14193197717314950975785972951
