L(s) = 1 | + (2.05 + 1.18i)2-s + (1.81 + 3.14i)4-s + (−1.71 − 2.97i)5-s + 3.86i·8-s − 8.15i·10-s + (−0.271 − 0.156i)11-s + (5.09 − 2.94i)13-s + (−0.958 + 1.65i)16-s + 0.953·17-s − 1.26i·19-s + (6.23 − 10.7i)20-s + (−0.372 − 0.645i)22-s + (5.91 − 3.41i)23-s + (−3.40 + 5.89i)25-s + 13.9·26-s + ⋯ |
L(s) = 1 | + (1.45 + 0.838i)2-s + (0.907 + 1.57i)4-s + (−0.768 − 1.33i)5-s + 1.36i·8-s − 2.57i·10-s + (−0.0819 − 0.0473i)11-s + (1.41 − 0.816i)13-s + (−0.239 + 0.414i)16-s + 0.231·17-s − 0.289i·19-s + (1.39 − 2.41i)20-s + (−0.0794 − 0.137i)22-s + (1.23 − 0.711i)23-s + (−0.680 + 1.17i)25-s + 2.73·26-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 1323 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.999 - 0.0304i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(3.427730178\) |
\(L(\frac12)\) |
\(\approx\) |
\(3.427730178\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 \) |
good | 2 | \( 1 + (-2.05 - 1.18i)T + (1 + 1.73i)T^{2} \) |
| 5 | \( 1 + (1.71 + 2.97i)T + (-2.5 + 4.33i)T^{2} \) |
| 11 | \( 1 + (0.271 + 0.156i)T + (5.5 + 9.52i)T^{2} \) |
| 13 | \( 1 + (-5.09 + 2.94i)T + (6.5 - 11.2i)T^{2} \) |
| 17 | \( 1 - 0.953T + 17T^{2} \) |
| 19 | \( 1 + 1.26iT - 19T^{2} \) |
| 23 | \( 1 + (-5.91 + 3.41i)T + (11.5 - 19.9i)T^{2} \) |
| 29 | \( 1 + (3.43 + 1.98i)T + (14.5 + 25.1i)T^{2} \) |
| 31 | \( 1 + (4.53 - 2.61i)T + (15.5 - 26.8i)T^{2} \) |
| 37 | \( 1 - 5.37T + 37T^{2} \) |
| 41 | \( 1 + (-0.0699 - 0.121i)T + (-20.5 + 35.5i)T^{2} \) |
| 43 | \( 1 + (-1.44 + 2.49i)T + (-21.5 - 37.2i)T^{2} \) |
| 47 | \( 1 + (1.00 - 1.74i)T + (-23.5 - 40.7i)T^{2} \) |
| 53 | \( 1 - 11.9iT - 53T^{2} \) |
| 59 | \( 1 + (-0.824 - 1.42i)T + (-29.5 + 51.0i)T^{2} \) |
| 61 | \( 1 + (2.57 + 1.48i)T + (30.5 + 52.8i)T^{2} \) |
| 67 | \( 1 + (-0.934 - 1.61i)T + (-33.5 + 58.0i)T^{2} \) |
| 71 | \( 1 + 10.9iT - 71T^{2} \) |
| 73 | \( 1 - 0.409iT - 73T^{2} \) |
| 79 | \( 1 + (5.23 - 9.06i)T + (-39.5 - 68.4i)T^{2} \) |
| 83 | \( 1 + (4.00 - 6.92i)T + (-41.5 - 71.8i)T^{2} \) |
| 89 | \( 1 + 2.11T + 89T^{2} \) |
| 97 | \( 1 + (10.5 + 6.06i)T + (48.5 + 84.0i)T^{2} \) |
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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
−9.292498216464550271376914387698, −8.523721488636449736606147037960, −7.88855146012771712811920898721, −7.08968508458595559263618474931, −6.01241099735460698178746465636, −5.39514365820962008203184423163, −4.58811543106679342853204163744, −3.90084363496922742905655839021, −3.01296501339555945083572010339, −0.980472668677057482904380744210,
1.60378410019753644818545653810, 2.83356480379910319205219408309, 3.59344600865108496999924724610, 4.07619015535051510430254484278, 5.28949841335105300223520258342, 6.18488138239410890751413323804, 6.88820649852555839507550101095, 7.76591445826935645728218853132, 8.934921868785049476846846527520, 10.04862968199085635886533080905