| L(s) = 1 | + 2·2-s + 2·4-s + 5-s − 3·7-s + 2·10-s + 2·11-s − 5·13-s − 6·14-s − 4·16-s + 8·17-s + 19-s + 2·20-s + 4·22-s − 6·23-s + 25-s − 10·26-s − 6·28-s − 2·29-s − 8·32-s + 16·34-s − 3·35-s + 5·37-s + 2·38-s + 10·41-s + 4·43-s + 4·44-s − 12·46-s + ⋯ |
| L(s) = 1 | + 1.41·2-s + 4-s + 0.447·5-s − 1.13·7-s + 0.632·10-s + 0.603·11-s − 1.38·13-s − 1.60·14-s − 16-s + 1.94·17-s + 0.229·19-s + 0.447·20-s + 0.852·22-s − 1.25·23-s + 1/5·25-s − 1.96·26-s − 1.13·28-s − 0.371·29-s − 1.41·32-s + 2.74·34-s − 0.507·35-s + 0.821·37-s + 0.324·38-s + 1.56·41-s + 0.609·43-s + 0.603·44-s − 1.76·46-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 135 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.966671309\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.966671309\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 \) |
| 5 | \( 1 - T \) |
| good | 2 | \( 1 - p T + p T^{2} \) |
| 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 + 5 T + p T^{2} \) |
| 17 | \( 1 - 8 T + p T^{2} \) |
| 19 | \( 1 - T + p T^{2} \) |
| 23 | \( 1 + 6 T + p T^{2} \) |
| 29 | \( 1 + 2 T + p T^{2} \) |
| 31 | \( 1 + p T^{2} \) |
| 37 | \( 1 - 5 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 + 4 T + p T^{2} \) |
| 53 | \( 1 - 2 T + p T^{2} \) |
| 59 | \( 1 - 8 T + p T^{2} \) |
| 61 | \( 1 - 7 T + p T^{2} \) |
| 67 | \( 1 + 9 T + p T^{2} \) |
| 71 | \( 1 + 2 T + p T^{2} \) |
| 73 | \( 1 + 5 T + p T^{2} \) |
| 79 | \( 1 + 3 T + p T^{2} \) |
| 83 | \( 1 + 6 T + p T^{2} \) |
| 89 | \( 1 - 12 T + p T^{2} \) |
| 97 | \( 1 + 13 T + p T^{2} \) |
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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
−13.16883187435091588893861858663, −12.42586889673726624358472238396, −11.78160851319161857295959451588, −10.03955482032079710811421473575, −9.412404196511086823579961082776, −7.49513099271809325678009295871, −6.25177698744351379012073167484, −5.43765056456230093289829322534, −3.97301637909484367451284003600, −2.76209099156017414000056665621,
2.76209099156017414000056665621, 3.97301637909484367451284003600, 5.43765056456230093289829322534, 6.25177698744351379012073167484, 7.49513099271809325678009295871, 9.412404196511086823579961082776, 10.03955482032079710811421473575, 11.78160851319161857295959451588, 12.42586889673726624358472238396, 13.16883187435091588893861858663
