L(s) = 1 | + (0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 1.73i·19-s + (−0.5 − 0.866i)25-s + 0.999·28-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (−0.499 − 0.866i)52-s + 1.73i·61-s − 0.999·64-s + 2·67-s + (−1.5 + 0.866i)73-s + ⋯ |
L(s) = 1 | + (0.5 − 0.866i)4-s + (0.5 + 0.866i)7-s + (0.5 − 0.866i)13-s + (−0.499 − 0.866i)16-s + 1.73i·19-s + (−0.5 − 0.866i)25-s + 0.999·28-s + (−0.5 − 0.866i)37-s + (−0.5 − 0.866i)43-s + (−0.499 + 0.866i)49-s + (−0.499 − 0.866i)52-s + 1.73i·61-s − 0.999·64-s + 2·67-s + (−1.5 + 0.866i)73-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 819 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.923 + 0.384i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.141167590\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.141167590\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 3 | \( 1 \) |
| 7 | \( 1 + (-0.5 - 0.866i)T \) |
| 13 | \( 1 + (-0.5 + 0.866i)T \) |
good | 2 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 5 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 11 | \( 1 + T^{2} \) |
| 17 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 19 | \( 1 - 1.73iT - T^{2} \) |
| 23 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 29 | \( 1 + (-0.5 - 0.866i)T^{2} \) |
| 31 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 37 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 41 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 43 | \( 1 + (0.5 + 0.866i)T + (-0.5 + 0.866i)T^{2} \) |
| 47 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 53 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 59 | \( 1 + (0.5 + 0.866i)T^{2} \) |
| 61 | \( 1 - 1.73iT - T^{2} \) |
| 67 | \( 1 - 2T + T^{2} \) |
| 71 | \( 1 + (-0.5 + 0.866i)T^{2} \) |
| 73 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
| 79 | \( 1 + (1 - 1.73i)T + (-0.5 - 0.866i)T^{2} \) |
| 83 | \( 1 - T^{2} \) |
| 89 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 97 | \( 1 + (1.5 - 0.866i)T + (0.5 - 0.866i)T^{2} \) |
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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
−10.36651498786995526246140185940, −9.745529986161441691621849835214, −8.603685693725296909688919642642, −7.965146770870713980278781464670, −6.81686260175457226496083490539, −5.66816164337314419103386480039, −5.54769636772491087569489348714, −4.03204835574637248451053451884, −2.62269463602989264999895534160, −1.53086773007200169560309015155,
1.71960731527707592749539533693, 3.08883804952070340749989553130, 4.08310327443642008501866969417, 4.95538752550857059906230171658, 6.48854271412159783648826316568, 7.04899505417660130099545624888, 7.87579694828925260051585480360, 8.698949477041907137226865956737, 9.571791563743384835220266351617, 10.77273491722723459770137964034