L(s) = 1 | + (−0.978 − 0.611i)2-s + (−2.30 + 0.662i)3-s + (−0.293 − 0.601i)4-s + (2.15 − 0.608i)5-s + (2.66 + 0.764i)6-s + (0.629 − 1.09i)7-s + (−0.321 + 3.06i)8-s + (2.35 − 1.47i)9-s + (−2.47 − 0.719i)10-s + (0.499 − 0.555i)11-s + (1.07 + 1.19i)12-s + (0.209 + 5.98i)13-s + (−1.28 + 0.681i)14-s + (−4.56 + 2.83i)15-s + (1.36 − 1.74i)16-s + (−1.59 + 0.111i)17-s + ⋯ |
L(s) = 1 | + (−0.691 − 0.432i)2-s + (−1.33 + 0.382i)3-s + (−0.146 − 0.300i)4-s + (0.962 − 0.272i)5-s + (1.08 + 0.311i)6-s + (0.237 − 0.412i)7-s + (−0.113 + 1.08i)8-s + (0.784 − 0.490i)9-s + (−0.783 − 0.227i)10-s + (0.150 − 0.167i)11-s + (0.310 + 0.345i)12-s + (0.0579 + 1.66i)13-s + (−0.342 + 0.182i)14-s + (−1.17 + 0.731i)15-s + (0.340 − 0.436i)16-s + (−0.386 + 0.0270i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 475 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.535 + 0.844i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(0.591254 - 0.325395i\) |
\(L(\frac12)\) |
\(\approx\) |
\(0.591254 - 0.325395i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 5 | \( 1 + (-2.15 + 0.608i)T \) |
| 19 | \( 1 + (1.77 + 3.98i)T \) |
good | 2 | \( 1 + (0.978 + 0.611i)T + (0.876 + 1.79i)T^{2} \) |
| 3 | \( 1 + (2.30 - 0.662i)T + (2.54 - 1.58i)T^{2} \) |
| 7 | \( 1 + (-0.629 + 1.09i)T + (-3.5 - 6.06i)T^{2} \) |
| 11 | \( 1 + (-0.499 + 0.555i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (-0.209 - 5.98i)T + (-12.9 + 0.906i)T^{2} \) |
| 17 | \( 1 + (1.59 - 0.111i)T + (16.8 - 2.36i)T^{2} \) |
| 23 | \( 1 + (-7.02 + 0.987i)T + (22.1 - 6.33i)T^{2} \) |
| 29 | \( 1 + (0.0907 + 0.00634i)T + (28.7 + 4.03i)T^{2} \) |
| 31 | \( 1 + (-5.43 + 2.42i)T + (20.7 - 23.0i)T^{2} \) |
| 37 | \( 1 + (0.862 - 2.65i)T + (-29.9 - 21.7i)T^{2} \) |
| 41 | \( 1 + (-4.92 + 6.30i)T + (-9.91 - 39.7i)T^{2} \) |
| 43 | \( 1 + (1.24 + 7.03i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (-0.369 - 0.0258i)T + (46.5 + 6.54i)T^{2} \) |
| 53 | \( 1 + (4.48 + 9.20i)T + (-32.6 + 41.7i)T^{2} \) |
| 59 | \( 1 + (1.73 + 4.30i)T + (-42.4 + 40.9i)T^{2} \) |
| 61 | \( 1 + (-3.48 + 0.490i)T + (58.6 - 16.8i)T^{2} \) |
| 67 | \( 1 + (-1.28 - 5.14i)T + (-59.1 + 31.4i)T^{2} \) |
| 71 | \( 1 + (7.40 + 7.15i)T + (2.47 + 70.9i)T^{2} \) |
| 73 | \( 1 + (0.0977 - 2.79i)T + (-72.8 - 5.09i)T^{2} \) |
| 79 | \( 1 + (-15.2 + 4.37i)T + (66.9 - 41.8i)T^{2} \) |
| 83 | \( 1 + (-0.0853 + 0.0379i)T + (55.5 - 61.6i)T^{2} \) |
| 89 | \( 1 + (-6.15 - 7.87i)T + (-21.5 + 86.3i)T^{2} \) |
| 97 | \( 1 + (1.83 - 7.35i)T + (-85.6 - 45.5i)T^{2} \) |
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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
−10.94288178734817827546011998892, −10.08850214774102903775877610340, −9.254752661230362528768547287277, −8.682417108934254420769837285866, −6.87098176943730456009279699912, −6.16983024105780122024598295104, −5.06594152108923774036700170744, −4.51810219696228954145225542210, −2.16429509201687780418279823025, −0.814715820187990548939709328307,
1.06932534262609362367244280355, 3.00113842644595276842882094960, 4.84581171649317740088775900603, 5.81327001961978903520729864164, 6.45449310120034869946123091285, 7.41329381330140920415920929888, 8.395506800117782151049684025515, 9.373419206907832756091269900105, 10.32605183575080318305002815350, 10.93493965277187346748345785085