L(s) = 1 | + (−1.48 − 2.57i)5-s + (4.09 + 2.36i)11-s + (3.54 + 2.04i)13-s + (−0.835 − 1.44i)17-s + (4.25 + 2.45i)19-s + (−4.25 + 2.45i)23-s + (−1.91 + 3.30i)25-s + (−0.238 + 0.137i)29-s − 1.60i·31-s + (−1.69 + 2.93i)37-s + (−3.55 + 6.15i)41-s + (5.22 + 9.05i)43-s − 10.9·47-s + (−0.707 + 0.408i)53-s − 14.0i·55-s + ⋯ |
L(s) = 1 | + (−0.664 − 1.15i)5-s + (1.23 + 0.712i)11-s + (0.981 + 0.566i)13-s + (−0.202 − 0.350i)17-s + (0.975 + 0.563i)19-s + (−0.886 + 0.511i)23-s + (−0.382 + 0.661i)25-s + (−0.0442 + 0.0255i)29-s − 0.287i·31-s + (−0.278 + 0.483i)37-s + (−0.555 + 0.961i)41-s + (0.797 + 1.38i)43-s − 1.60·47-s + (−0.0971 + 0.0560i)53-s − 1.89i·55-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 5292 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.620 - 0.783i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.572072885\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.572072885\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 5 | \( 1 + (1.48 + 2.57i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (-4.09 - 2.36i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-3.54 - 2.04i)T + (6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + (0.835 + 1.44i)T + (-8.5 + 14.7i)T^{2} \) |
| 19 | \( 1 + (-4.25 - 2.45i)T + (9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (4.25 - 2.45i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (0.238 - 0.137i)T + (14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + 1.60iT - 31T^{2} \) |
| 37 | \( 1 + (1.69 - 2.93i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (3.55 - 6.15i)T + (-20.5 - 35.5i)T^{2} \) |
| 43 | \( 1 + (-5.22 - 9.05i)T + (-21.5 + 37.2i)T^{2} \) |
| 47 | \( 1 + 10.9T + 47T^{2} \) |
| 53 | \( 1 + (0.707 - 0.408i)T + (26.5 - 45.8i)T^{2} \) |
| 59 | \( 1 - 2.74T + 59T^{2} \) |
| 61 | \( 1 - 7.20iT - 61T^{2} \) |
| 67 | \( 1 - 11.6T + 67T^{2} \) |
| 71 | \( 1 - 10.4iT - 71T^{2} \) |
| 73 | \( 1 + (13.6 - 7.88i)T + (36.5 - 63.2i)T^{2} \) |
| 79 | \( 1 + 12.3T + 79T^{2} \) |
| 83 | \( 1 + (4.03 + 6.99i)T + (-41.5 + 71.8i)T^{2} \) |
| 89 | \( 1 + (-4.60 + 7.98i)T + (-44.5 - 77.0i)T^{2} \) |
| 97 | \( 1 + (7.00 - 4.04i)T + (48.5 - 84.0i)T^{2} \) |
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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
−8.349308004253430213522583443267, −7.69531950599190000207953790030, −6.87785194663773559886546162841, −6.16193043420721432859605972147, −5.33006625617871189771113810712, −4.38110040006025614903211570357, −4.09885652934196833136079334907, −3.15986964283708393170316925955, −1.64383124950260564483747049732, −1.13155235150422696965067720946,
0.47164921621660950351727199903, 1.73139684338361237408258065947, 2.97190091062950452058188944894, 3.58374007715243742053177357410, 4.05407807208525349788617678724, 5.30011765961042358228736484410, 6.10927791522086943498438106235, 6.67846567141362593231143010219, 7.27277136857433556663777721857, 8.115755934582645964327271496169