L(s) = 1 | + (−1.53 − 1.28i)4-s + (−3.06 + 2.57i)7-s + (6.57 − 2.39i)13-s + (0.694 + 3.93i)16-s + (0.5 + 0.866i)19-s + (4.69 + 1.71i)25-s + 8·28-s + (8.42 + 7.07i)31-s + (5 − 8.66i)37-s + (0.868 + 4.92i)43-s + (1.56 − 8.86i)49-s + (−13.1 − 4.78i)52-s + (−0.766 + 0.642i)61-s + (4.00 − 6.92i)64-s + (−4.69 + 1.71i)67-s + ⋯ |
L(s) = 1 | + (−0.766 − 0.642i)4-s + (−1.15 + 0.971i)7-s + (1.82 − 0.664i)13-s + (0.173 + 0.984i)16-s + (0.114 + 0.198i)19-s + (0.939 + 0.342i)25-s + 1.51·28-s + (1.51 + 1.26i)31-s + (0.821 − 1.42i)37-s + (0.132 + 0.750i)43-s + (0.223 − 1.26i)49-s + (−1.82 − 0.664i)52-s + (−0.0980 + 0.0823i)61-s + (0.500 − 0.866i)64-s + (−0.574 + 0.208i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 729 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.973 - 0.230i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(\approx\) |
\(1.13653 + 0.132842i\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.13653 + 0.132842i\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
good | 2 | \( 1 + (1.53 + 1.28i)T^{2} \) |
| 5 | \( 1 + (-4.69 - 1.71i)T^{2} \) |
| 7 | \( 1 + (3.06 - 2.57i)T + (1.21 - 6.89i)T^{2} \) |
| 11 | \( 1 + (-10.3 + 3.76i)T^{2} \) |
| 13 | \( 1 + (-6.57 + 2.39i)T + (9.95 - 8.35i)T^{2} \) |
| 17 | \( 1 + (-8.5 - 14.7i)T^{2} \) |
| 19 | \( 1 + (-0.5 - 0.866i)T + (-9.5 + 16.4i)T^{2} \) |
| 23 | \( 1 + (3.99 + 22.6i)T^{2} \) |
| 29 | \( 1 + (22.2 + 18.6i)T^{2} \) |
| 31 | \( 1 + (-8.42 - 7.07i)T + (5.38 + 30.5i)T^{2} \) |
| 37 | \( 1 + (-5 + 8.66i)T + (-18.5 - 32.0i)T^{2} \) |
| 41 | \( 1 + (31.4 - 26.3i)T^{2} \) |
| 43 | \( 1 + (-0.868 - 4.92i)T + (-40.4 + 14.7i)T^{2} \) |
| 47 | \( 1 + (8.16 - 46.2i)T^{2} \) |
| 53 | \( 1 + 53T^{2} \) |
| 59 | \( 1 + (-55.4 - 20.1i)T^{2} \) |
| 61 | \( 1 + (0.766 - 0.642i)T + (10.5 - 60.0i)T^{2} \) |
| 67 | \( 1 + (4.69 - 1.71i)T + (51.3 - 43.0i)T^{2} \) |
| 71 | \( 1 + (-35.5 - 61.4i)T^{2} \) |
| 73 | \( 1 + (-3.5 - 6.06i)T + (-36.5 + 63.2i)T^{2} \) |
| 79 | \( 1 + (-12.2 - 4.44i)T + (60.5 + 50.7i)T^{2} \) |
| 83 | \( 1 + (63.5 + 53.3i)T^{2} \) |
| 89 | \( 1 + (-44.5 + 77.0i)T^{2} \) |
| 97 | \( 1 + (-0.868 - 4.92i)T + (-91.1 + 33.1i)T^{2} \) |
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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
−10.37719194985201615781198833003, −9.446314362089780396250193881313, −8.860207680636327589962838723666, −8.152597809530267041362975953677, −6.54964488385060639079402672367, −5.99499604486112861398323860844, −5.18741081629584981848113567264, −3.86852740133365889913321318730, −2.88971687724150994353734800156, −1.05832123159129290306861063711,
0.823563746339314533382540274738, 3.02029054062207989109891076167, 3.85779063245516187222085309511, 4.58885160991286545714637689621, 6.14643423188643526032020318698, 6.77197118708064399064719947821, 7.88360060745879125646391381049, 8.678329946907627334241885756574, 9.472377073703819482909628472325, 10.23214464087526367844708396379