L(s) = 1 | + 1.55·2-s + (0.244 − 0.423i)3-s + 0.417·4-s + (0.595 − 1.03i)5-s + (0.380 − 0.658i)6-s + (−2.44 + 1.01i)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 + (2.86 − 2.19i)13-s + (−3.79 + 1.58i)14-s + (−0.291 − 0.504i)15-s − 4.66·16-s − 0.906·17-s + ⋯ |
L(s) = 1 | + 1.09·2-s + (0.141 − 0.244i)3-s + 0.208·4-s + (0.266 − 0.461i)5-s + (0.155 − 0.268i)6-s + (−0.922 + 0.385i)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 + (0.793 − 0.608i)13-s + (−1.01 + 0.423i)14-s + (−0.0752 − 0.130i)15-s − 1.16·16-s − 0.219·17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.982 + 0.188i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.48303 - 0.140838i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.48303 - 0.140838i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 7 | \( 1 + (2.44 - 1.01i)T \) |
| 13 | \( 1 + (-2.86 + 2.19i)T \) |
good | 2 | \( 1 - 1.55T + 2T^{2} \) |
| 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 | \( 1 + 0.906T + 17T^{2} \) |
| 19 | \( 1 + (3.34 + 5.79i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 - 3.59T + 23T^{2} \) |
| 29 | \( 1 + (4.25 + 7.37i)T + (-14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (-2.64 - 4.57i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 - 4.99T + 37T^{2} \) |
| 41 | \( 1 + (0.768 + 1.33i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (2.71 - 4.70i)T + (-21.5 - 37.2i)T^{2} \) |
| 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 | \( 1 + 10.2T + 59T^{2} \) |
| 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 | \( 1 + (-1.26 + 2.19i)T + (-35.5 - 61.4i)T^{2} \) |
| 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 | \( 1 + 17.7T + 89T^{2} \) |
| 97 | \( 1 + (3.10 - 5.37i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−13.54117812416410673344257849813, −13.14829077539790113206673103600, −12.56291395049297522078005595185, −11.05781898383187491507186549384, −9.624762272587006696403208114707, −8.559995177069430509793327840878, −6.87434185488195645711013659510, −5.60397517133561060689009966916, −4.47578615373994744792370653383, −2.75702149941998175409532461712,
3.22984496892227500874324299375, 4.16169913094958830987836304505, 5.96743247130696181165604926837, 6.72573767094484853686076855630, 8.751211304553328590149226388240, 9.821564504447169770855584111387, 10.99964022217774494544446391386, 12.43005967859820526229938118632, 13.13661580884291146824312754955, 14.07883479931745162391450408983