| L(s) = 1 | − 11·7-s − 23·13-s − 37·19-s − 46·31-s + 73·37-s + 22·43-s + 72·49-s + 47·61-s + 13·67-s − 143·73-s + 11·79-s + 253·91-s + 169·97-s + 157·103-s − 214·109-s + ⋯ |
| L(s) = 1 | − 1.57·7-s − 1.76·13-s − 1.94·19-s − 1.48·31-s + 1.97·37-s + 0.511·43-s + 1.46·49-s + 0.770·61-s + 0.194·67-s − 1.95·73-s + 0.139·79-s + 2.78·91-s + 1.74·97-s + 1.52·103-s − 1.96·109-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(3-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2700 ^{s/2} \, \Gamma_{\C}(s+1) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(\frac{3}{2})\) |
\(\approx\) |
\(0.6320564949\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.6320564949\) |
| \(L(2)\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| 5 | \( 1 \) |
| good | 7 | \( 1 + 11 T + p^{2} T^{2} \) |
| 11 | \( ( 1 - p T )( 1 + p T ) \) |
| 13 | \( 1 + 23 T + p^{2} T^{2} \) |
| 17 | \( ( 1 - p T )( 1 + p T ) \) |
| 19 | \( 1 + 37 T + p^{2} T^{2} \) |
| 23 | \( ( 1 - p T )( 1 + p T ) \) |
| 29 | \( ( 1 - p T )( 1 + p T ) \) |
| 31 | \( 1 + 46 T + p^{2} T^{2} \) |
| 37 | \( 1 - 73 T + p^{2} T^{2} \) |
| 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 | \( 1 - 47 T + p^{2} T^{2} \) |
| 67 | \( 1 - 13 T + p^{2} T^{2} \) |
| 71 | \( ( 1 - p T )( 1 + p T ) \) |
| 73 | \( 1 + 143 T + p^{2} T^{2} \) |
| 79 | \( 1 - 11 T + p^{2} T^{2} \) |
| 83 | \( ( 1 - p T )( 1 + p T ) \) |
| 89 | \( ( 1 - p T )( 1 + p T ) \) |
| 97 | \( 1 - 169 T + p^{2} T^{2} \) |
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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
−8.859628891705010286076721173424, −7.77071152683731977707348548484, −7.09431720085201131093085805398, −6.39899877728446226612839892253, −5.72350165526389847005895148721, −4.63698726015282828895463099102, −3.88506689913411736536972370708, −2.82272241636683467078495862519, −2.16249139217215029014371662607, −0.36864385809697591827528920883,
0.36864385809697591827528920883, 2.16249139217215029014371662607, 2.82272241636683467078495862519, 3.88506689913411736536972370708, 4.63698726015282828895463099102, 5.72350165526389847005895148721, 6.39899877728446226612839892253, 7.09431720085201131093085805398, 7.77071152683731977707348548484, 8.859628891705010286076721173424
