| L(s) = 1 | + 2-s + 4-s + 4·5-s + 3·7-s + 8-s + 4·10-s − 5·11-s + 3·13-s + 3·14-s + 16-s − 3·17-s − 7·19-s + 4·20-s − 5·22-s − 9·23-s + 11·25-s + 3·26-s + 3·28-s − 2·31-s + 32-s − 3·34-s + 12·35-s + 37-s − 7·38-s + 4·40-s − 6·41-s + 4·43-s + ⋯ |
| L(s) = 1 | + 0.707·2-s + 1/2·4-s + 1.78·5-s + 1.13·7-s + 0.353·8-s + 1.26·10-s − 1.50·11-s + 0.832·13-s + 0.801·14-s + 1/4·16-s − 0.727·17-s − 1.60·19-s + 0.894·20-s − 1.06·22-s − 1.87·23-s + 11/5·25-s + 0.588·26-s + 0.566·28-s − 0.359·31-s + 0.176·32-s − 0.514·34-s + 2.02·35-s + 0.164·37-s − 1.13·38-s + 0.632·40-s − 0.937·41-s + 0.609·43-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 666 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(3.068028744\) |
| \(L(\frac12)\) |
\(\approx\) |
\(3.068028744\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 - T \) |
| 3 | \( 1 \) |
| 37 | \( 1 - T \) |
| good | 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 - 3 T + p T^{2} \) |
| 11 | \( 1 + 5 T + p T^{2} \) |
| 13 | \( 1 - 3 T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 7 T + p T^{2} \) |
| 23 | \( 1 + 9 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 2 T + p T^{2} \) |
| 41 | \( 1 + 6 T + p T^{2} \) |
| 43 | \( 1 - 4 T + p T^{2} \) |
| 47 | \( 1 - 10 T + p T^{2} \) |
| 53 | \( 1 + 3 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 + 2 T + p T^{2} \) |
| 67 | \( 1 - 6 T + p T^{2} \) |
| 71 | \( 1 - 12 T + p T^{2} \) |
| 73 | \( 1 - 13 T + p T^{2} \) |
| 79 | \( 1 + 6 T + p T^{2} \) |
| 83 | \( 1 + 5 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 - 6 T + p T^{2} \) |
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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
−10.66123771129793013895319012928, −9.914305521533027310416702610710, −8.668480330778590791299654356016, −7.989549022432806984304290222280, −6.61856061275031960976062588393, −5.83968165780663709630491739946, −5.19848296413414906626093432671, −4.18804468941342822592597020971, −2.40265859153889095661502631676, −1.88395043108710714192663999988,
1.88395043108710714192663999988, 2.40265859153889095661502631676, 4.18804468941342822592597020971, 5.19848296413414906626093432671, 5.83968165780663709630491739946, 6.61856061275031960976062588393, 7.989549022432806984304290222280, 8.668480330778590791299654356016, 9.914305521533027310416702610710, 10.66123771129793013895319012928
