L(s) = 1 | + (0.0581 − 0.998i)3-s + (−0.993 − 0.116i)9-s + (0.512 − 0.257i)11-s + (0.606 − 0.509i)17-s + (−1.36 − 1.14i)19-s + (0.396 − 0.918i)25-s + (−0.173 + 0.984i)27-s + (−0.227 − 0.526i)33-s + (−0.0694 + 0.0932i)41-s + (0.0694 − 1.19i)43-s + (−0.0581 − 0.998i)49-s + (−0.473 − 0.635i)51-s + (−1.22 + 1.30i)57-s + (1.49 + 0.749i)59-s + (−1.36 − 0.159i)67-s + ⋯ |
L(s) = 1 | + (0.0581 − 0.998i)3-s + (−0.993 − 0.116i)9-s + (0.512 − 0.257i)11-s + (0.606 − 0.509i)17-s + (−1.36 − 1.14i)19-s + (0.396 − 0.918i)25-s + (−0.173 + 0.984i)27-s + (−0.227 − 0.526i)33-s + (−0.0694 + 0.0932i)41-s + (0.0694 − 1.19i)43-s + (−0.0581 − 0.998i)49-s + (−0.473 − 0.635i)51-s + (−1.22 + 1.30i)57-s + (1.49 + 0.749i)59-s + (−1.36 − 0.159i)67-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (-0.360 + 0.932i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
\(L(\frac{1}{2})\) |
\(\approx\) |
\(1.093355170\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.093355170\) |
\(L(1)\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 3 | \( 1 + (-0.0581 + 0.998i)T \) |
good | 5 | \( 1 + (-0.396 + 0.918i)T^{2} \) |
| 7 | \( 1 + (0.0581 + 0.998i)T^{2} \) |
| 11 | \( 1 + (-0.512 + 0.257i)T + (0.597 - 0.802i)T^{2} \) |
| 13 | \( 1 + (0.686 - 0.727i)T^{2} \) |
| 17 | \( 1 + (-0.606 + 0.509i)T + (0.173 - 0.984i)T^{2} \) |
| 19 | \( 1 + (1.36 + 1.14i)T + (0.173 + 0.984i)T^{2} \) |
| 23 | \( 1 + (0.0581 - 0.998i)T^{2} \) |
| 29 | \( 1 + (-0.973 + 0.230i)T^{2} \) |
| 31 | \( 1 + (-0.893 + 0.448i)T^{2} \) |
| 37 | \( 1 + (0.939 - 0.342i)T^{2} \) |
| 41 | \( 1 + (0.0694 - 0.0932i)T + (-0.286 - 0.957i)T^{2} \) |
| 43 | \( 1 + (-0.0694 + 1.19i)T + (-0.993 - 0.116i)T^{2} \) |
| 47 | \( 1 + (-0.893 - 0.448i)T^{2} \) |
| 53 | \( 1 + (0.5 - 0.866i)T^{2} \) |
| 59 | \( 1 + (-1.49 - 0.749i)T + (0.597 + 0.802i)T^{2} \) |
| 61 | \( 1 + (0.835 - 0.549i)T^{2} \) |
| 67 | \( 1 + (1.36 + 0.159i)T + (0.973 + 0.230i)T^{2} \) |
| 71 | \( 1 + (-0.766 - 0.642i)T^{2} \) |
| 73 | \( 1 + (-1.86 + 0.679i)T + (0.766 - 0.642i)T^{2} \) |
| 79 | \( 1 + (0.286 - 0.957i)T^{2} \) |
| 83 | \( 1 + (0.207 + 0.278i)T + (-0.286 + 0.957i)T^{2} \) |
| 89 | \( 1 + (1.43 - 0.524i)T + (0.766 - 0.642i)T^{2} \) |
| 97 | \( 1 + (1.62 + 1.06i)T + (0.396 + 0.918i)T^{2} \) |
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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
−8.657133263895251515132185532640, −8.170969566948614450032692426850, −7.11221561619176325646487938161, −6.71559172177529225292265720247, −5.89571551627122371420062667942, −5.01306507663102748662740817360, −3.96047613101445157795766612847, −2.86586714759134692146189568969, −2.05287816733723600769647254039, −0.72090489308195568551623401066,
1.63083498498791637130970180426, 2.89964744644242775316032688290, 3.84105893832032263948564883079, 4.38326386250914940531706650616, 5.41851386374420909962467915842, 6.06299393958797455643168659870, 6.94429638108302266226993898190, 8.090546181671619063883237065814, 8.492469104090376150610010197706, 9.497039141185297856847730715494