L(s) = 1 | + 4·4-s − 13·7-s − 13-s + 16·16-s + 11·19-s + 25·25-s − 52·28-s − 46·31-s + 47·37-s − 22·43-s + 120·49-s − 4·52-s − 121·61-s + 64·64-s − 109·67-s − 97·73-s + 44·76-s + 131·79-s + 13·91-s + 167·97-s + 100·100-s − 37·103-s − 214·109-s − 208·112-s + ⋯ |
L(s) = 1 | + 4-s − 1.85·7-s − 0.0769·13-s + 16-s + 0.578·19-s + 25-s − 1.85·28-s − 1.48·31-s + 1.27·37-s − 0.511·43-s + 2.44·49-s − 0.0769·52-s − 1.98·61-s + 64-s − 1.62·67-s − 1.32·73-s + 0.578·76-s + 1.65·79-s + 1/7·91-s + 1.72·97-s + 100-s − 0.359·103-s − 1.96·109-s − 1.85·112-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 27 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(\frac{3}{2})\) |
\(\approx\) |
\(1.000965941\) |
\(L(\frac12)\) |
\(\approx\) |
\(1.000965941\) |
\(L(2)\) |
|
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 | \( ( 1 - p T )( 1 + p T ) \) |
| 5 | \( ( 1 - p T )( 1 + p T ) \) |
| 7 | \( 1 + 13 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 - 11 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 46 T + p^{2} T^{2} \) |
| 37 | \( 1 - 47 T + p^{2} T^{2} \) |
| 41 | \( ( 1 - p T )( 1 + p T ) \) |
| 43 | \( 1 + 22 T + p^{2} T^{2} \) |
| 47 | \( ( 1 - p T )( 1 + p T ) \) |
| 53 | \( ( 1 - p T )( 1 + p T ) \) |
| 59 | \( ( 1 - p T )( 1 + p T ) \) |
| 61 | \( 1 + 121 T + p^{2} T^{2} \) |
| 67 | \( 1 + 109 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 97 T + p^{2} T^{2} \) |
| 79 | \( 1 - 131 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 167 T + p^{2} T^{2} \) |
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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
−16.66700335745736228791898170610, −16.12719347488778567737906039955, −14.93317694987577442968540511904, −13.18826623214618288315647120749, −12.16309481374790876133361323529, −10.64354618326625880843609971954, −9.385096739194486683174830096085, −7.25050227093460265001191315374, −6.08002078378355951356788956969, −3.12785755550327188554888192988,
3.12785755550327188554888192988, 6.08002078378355951356788956969, 7.25050227093460265001191315374, 9.385096739194486683174830096085, 10.64354618326625880843609971954, 12.16309481374790876133361323529, 13.18826623214618288315647120749, 14.93317694987577442968540511904, 16.12719347488778567737906039955, 16.66700335745736228791898170610