L(s) = 1 | + (0.551 − 0.955i)2-s + (−1.67 − 0.441i)3-s + (0.391 + 0.678i)4-s + (0.0527 + 0.0913i)5-s + (−1.34 + 1.35i)6-s + 3.07·8-s + (2.60 + 1.47i)9-s + 0.116·10-s + (−1.66 + 2.89i)11-s + (−0.356 − 1.30i)12-s + (1.23 + 2.14i)13-s + (−0.0479 − 0.176i)15-s + (0.909 − 1.57i)16-s − 1.61·17-s + (2.85 − 1.67i)18-s + 7.68·19-s + ⋯ |
L(s) = 1 | + (0.389 − 0.675i)2-s + (−0.966 − 0.255i)3-s + (0.195 + 0.339i)4-s + (0.0235 + 0.0408i)5-s + (−0.549 + 0.553i)6-s + 1.08·8-s + (0.869 + 0.493i)9-s + 0.0367·10-s + (−0.503 + 0.871i)11-s + (−0.102 − 0.378i)12-s + (0.343 + 0.595i)13-s + (−0.0123 − 0.0455i)15-s + (0.227 − 0.393i)16-s − 0.391·17-s + (0.672 − 0.395i)18-s + 1.76·19-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 441 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.986 + 0.166i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.45082 - 0.121287i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.45082 - 0.121287i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 + (1.67 + 0.441i)T \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-0.551 + 0.955i)T + (-1 - 1.73i)T^{2} \) |
| 5 | \( 1 + (-0.0527 - 0.0913i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (1.66 - 2.89i)T + (-5.5 - 9.52i)T^{2} \) |
| 13 | \( 1 + (-1.23 - 2.14i)T + (-6.5 + 11.2i)T^{2} \) |
| 17 | \( 1 + 1.61T + 17T^{2} \) |
| 19 | \( 1 - 7.68T + 19T^{2} \) |
| 23 | \( 1 + (-0.948 - 1.64i)T + (-11.5 + 19.9i)T^{2} \) |
| 29 | \( 1 + (-4.64 + 8.04i)T + (-14.5 - 25.1i)T^{2} \) |
| 31 | \( 1 + (-4.63 - 8.02i)T + (-15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + 1.98T + 37T^{2} \) |
| 41 | \( 1 + (3.74 + 6.48i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (3.77 - 6.53i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.59 - 2.76i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 + 9.97T + 53T^{2} \) |
| 59 | \( 1 + (-2.22 - 3.86i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.83 - 4.91i)T + (-30.5 - 52.8i)T^{2} \) |
| 67 | \( 1 + (4.98 + 8.63i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 - 3.29T + 71T^{2} \) |
| 73 | \( 1 - 4.72T + 73T^{2} \) |
| 79 | \( 1 + (3.84 - 6.66i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (-0.584 + 1.01i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 6.02T + 89T^{2} \) |
| 97 | \( 1 + (-1.90 + 3.29i)T + (-48.5 - 84.0i)T^{2} \) |
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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
−11.32506577611526247797564276217, −10.44145692439900791221572583273, −9.725937093845909830484652582257, −8.162964195501716387699103589792, −7.24569052719994498786223310542, −6.48838355619880009931429991899, −5.07333652283792946405091584887, −4.36527735313029493139939217340, −2.88621225477121044351092735901, −1.50948830560387669301440078546,
1.08637519871768392925950430588, 3.30900031645613801471055842742, 4.88537802414448571713307736266, 5.39983609869076036588736022474, 6.30408324044949598339033360202, 7.13090511822536748905605504196, 8.179801246875899826080383286991, 9.539438895332028165422713006229, 10.45532279464124650496813008043, 11.07029306681072478075835030587