L(s) = 1 | + (−0.777 + 1.34i)2-s + (0.244 + 0.423i)3-s + (−0.208 − 0.361i)4-s + (0.595 − 1.03i)5-s − 0.760·6-s + (2.10 + 1.60i)7-s − 2.46·8-s + (1.38 − 2.39i)9-s + (0.926 + 1.60i)10-s + (−1.05 − 1.83i)11-s + (0.102 − 0.176i)12-s + (−3.79 + 1.58i)14-s + 0.582·15-s + (2.33 − 4.03i)16-s + (0.453 + 0.784i)17-s + (2.14 + 3.71i)18-s + ⋯ |
L(s) = 1 | + (−0.549 + 0.952i)2-s + (0.141 + 0.244i)3-s + (−0.104 − 0.180i)4-s + (0.266 − 0.461i)5-s − 0.310·6-s + (0.795 + 0.606i)7-s − 0.870·8-s + (0.460 − 0.796i)9-s + (0.292 + 0.507i)10-s + (−0.319 − 0.552i)11-s + (0.0294 − 0.0510i)12-s + (−1.01 + 0.423i)14-s + 0.150·15-s + (0.582 − 1.00i)16-s + (0.109 + 0.190i)17-s + (0.505 + 0.876i)18-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1183 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (-0.443 - 0.896i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.437887489\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.437887489\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (-2.10 - 1.60i)T \) |
| 13 | \( 1 \) |
good | 2 | \( 1 + (0.777 - 1.34i)T + (-1 - 1.73i)T^{2} \) |
| 3 | \( 1 + (-0.244 - 0.423i)T + (-1.5 + 2.59i)T^{2} \) |
| 5 | \( 1 + (-0.595 + 1.03i)T + (-2.5 - 4.33i)T^{2} \) |
| 11 | \( 1 + (1.05 + 1.83i)T + (-5.5 + 9.52i)T^{2} \) |
| 17 | \( 1 + (-0.453 - 0.784i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (3.34 - 5.79i)T + (-9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (1.79 - 3.11i)T + (-11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 - 8.51T + 29T^{2} \) |
| 31 | \( 1 + (-2.64 - 4.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (2.49 - 4.32i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 - 1.53T + 41T^{2} \) |
| 43 | \( 1 - 5.43T + 43T^{2} \) |
| 47 | \( 1 + (-1.59 + 2.75i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + (-1.41 - 2.44i)T + (-26.5 + 45.8i)T^{2} \) |
| 59 | \( 1 + (-5.12 - 8.87i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (-4.13 + 7.16i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (-1.87 - 3.24i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 2.53T + 71T^{2} \) |
| 73 | \( 1 + (-2.86 - 4.96i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (3.03 - 5.25i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + 11.6T + 83T^{2} \) |
| 89 | \( 1 + (-8.87 + 15.3i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 - 6.20T + 97T^{2} \) |
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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
−9.798001572002572433491862797666, −8.810951372716458058627980341820, −8.515997771732731506907511370991, −7.75996109071853766327905214686, −6.71926780856508143980993166984, −5.93924446589405157968792967343, −5.21641663622826491105915150270, −4.01377315108381843078063446578, −2.85446628604575294612051890057, −1.29710065117532689406433266524,
0.822757818520953141613265308000, 2.18393441743156440932815716528, 2.58637762954337047507637185959, 4.22973670597598125700543178510, 4.99933762781294089238381086071, 6.36003207325187467152626314703, 7.09888070048098231105412617205, 8.033195811273706876104670465245, 8.745186962151157353939320494182, 9.814038272433641205307006608492