| L(s) = 1 | + (0.284 + 0.206i)2-s + (−0.302 − 2.87i)3-s + (−0.579 − 1.78i)4-s + (0.508 − 0.880i)6-s + (−1.05 − 0.224i)7-s + (0.420 − 1.29i)8-s + (−5.25 + 1.11i)9-s + (−1.62 + 1.80i)11-s + (−4.95 + 2.20i)12-s + (−2.62 − 1.16i)13-s + (−0.254 − 0.282i)14-s + (−2.64 + 1.92i)16-s + (1.22 + 1.35i)17-s + (−1.72 − 0.767i)18-s + (1.93 − 0.861i)19-s + ⋯ |
| L(s) = 1 | + (0.201 + 0.146i)2-s + (−0.174 − 1.66i)3-s + (−0.289 − 0.892i)4-s + (0.207 − 0.359i)6-s + (−0.400 − 0.0850i)7-s + (0.148 − 0.458i)8-s + (−1.75 + 0.372i)9-s + (−0.490 + 0.544i)11-s + (−1.43 + 0.637i)12-s + (−0.728 − 0.324i)13-s + (−0.0680 − 0.0755i)14-s + (−0.662 + 0.481i)16-s + (0.296 + 0.328i)17-s + (−0.406 − 0.180i)18-s + (0.443 − 0.197i)19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 775 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.743 - 0.669i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.256689 + 0.668731i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.256689 + 0.668731i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 5 | \( 1 \) |
| 31 | \( 1 + (-1.15 + 5.44i)T \) |
| good | 2 | \( 1 + (-0.284 - 0.206i)T + (0.618 + 1.90i)T^{2} \) |
| 3 | \( 1 + (0.302 + 2.87i)T + (-2.93 + 0.623i)T^{2} \) |
| 7 | \( 1 + (1.05 + 0.224i)T + (6.39 + 2.84i)T^{2} \) |
| 11 | \( 1 + (1.62 - 1.80i)T + (-1.14 - 10.9i)T^{2} \) |
| 13 | \( 1 + (2.62 + 1.16i)T + (8.69 + 9.66i)T^{2} \) |
| 17 | \( 1 + (-1.22 - 1.35i)T + (-1.77 + 16.9i)T^{2} \) |
| 19 | \( 1 + (-1.93 + 0.861i)T + (12.7 - 14.1i)T^{2} \) |
| 23 | \( 1 + (-0.136 + 0.420i)T + (-18.6 - 13.5i)T^{2} \) |
| 29 | \( 1 + (-2.55 - 1.85i)T + (8.96 + 27.5i)T^{2} \) |
| 37 | \( 1 + (-1.57 + 2.72i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (-0.726 + 6.90i)T + (-40.1 - 8.52i)T^{2} \) |
| 43 | \( 1 + (-7.68 + 3.42i)T + (28.7 - 31.9i)T^{2} \) |
| 47 | \( 1 + (6.44 - 4.67i)T + (14.5 - 44.6i)T^{2} \) |
| 53 | \( 1 + (4.86 - 1.03i)T + (48.4 - 21.5i)T^{2} \) |
| 59 | \( 1 + (-1.25 - 11.9i)T + (-57.7 + 12.2i)T^{2} \) |
| 61 | \( 1 + 14.4T + 61T^{2} \) |
| 67 | \( 1 + (3.21 + 5.57i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + (1.64 - 0.348i)T + (64.8 - 28.8i)T^{2} \) |
| 73 | \( 1 + (9.60 - 10.6i)T + (-7.63 - 72.6i)T^{2} \) |
| 79 | \( 1 + (2.30 + 2.56i)T + (-8.25 + 78.5i)T^{2} \) |
| 83 | \( 1 + (-1.34 + 12.7i)T + (-81.1 - 17.2i)T^{2} \) |
| 89 | \( 1 + (-0.698 - 2.14i)T + (-72.0 + 52.3i)T^{2} \) |
| 97 | \( 1 + (1.05 + 3.24i)T + (-78.4 + 57.0i)T^{2} \) |
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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
−9.839541786625683169387331945756, −8.875118502794572564294349765447, −7.66351529158830397233542843802, −7.22305724259646677688372889591, −6.21933348293595150836310492362, −5.62934768755100222433971200061, −4.54497460128500467378958483908, −2.78388124629231472409356391263, −1.63733248420322082206295677393, −0.34003600619018242039682059322,
2.87186416981903421654003591919, 3.41312065742156203761992779215, 4.58018442206123408823311715365, 5.04202497017541799180718220873, 6.23989173424089604679610685260, 7.61006390472462629484045211739, 8.456537561807103932610356992240, 9.377947258730891652085713756898, 9.847758128570873150866516661636, 10.80861568740725577954968069810
