| L(s) = 1 | − 5-s − 2·7-s + 2·11-s + 13-s − 3·17-s − 2·19-s + 6·23-s − 4·25-s − 29-s + 8·31-s + 2·35-s + 37-s + 2·41-s − 10·43-s + 4·47-s − 3·49-s + 10·53-s − 2·55-s + 4·59-s + 9·61-s − 65-s + 14·67-s + 10·71-s − 9·73-s − 4·77-s − 10·79-s + 12·83-s + ⋯ |
| L(s) = 1 | − 0.447·5-s − 0.755·7-s + 0.603·11-s + 0.277·13-s − 0.727·17-s − 0.458·19-s + 1.25·23-s − 4/5·25-s − 0.185·29-s + 1.43·31-s + 0.338·35-s + 0.164·37-s + 0.312·41-s − 1.52·43-s + 0.583·47-s − 3/7·49-s + 1.37·53-s − 0.269·55-s + 0.520·59-s + 1.15·61-s − 0.124·65-s + 1.71·67-s + 1.18·71-s − 1.05·73-s − 0.455·77-s − 1.12·79-s + 1.31·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 2592 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.398780138\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.398780138\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 \) |
| good | 5 | \( 1 + T + p T^{2} \) |
| 7 | \( 1 + 2 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - T + p T^{2} \) |
| 17 | \( 1 + 3 T + p T^{2} \) |
| 19 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 - 6 T + p T^{2} \) |
| 29 | \( 1 + T + p T^{2} \) |
| 31 | \( 1 - 8 T + p T^{2} \) |
| 37 | \( 1 - T + p T^{2} \) |
| 41 | \( 1 - 2 T + p T^{2} \) |
| 43 | \( 1 + 10 T + p T^{2} \) |
| 47 | \( 1 - 4 T + p T^{2} \) |
| 53 | \( 1 - 10 T + p T^{2} \) |
| 59 | \( 1 - 4 T + p T^{2} \) |
| 61 | \( 1 - 9 T + p T^{2} \) |
| 67 | \( 1 - 14 T + p T^{2} \) |
| 71 | \( 1 - 10 T + p T^{2} \) |
| 73 | \( 1 + 9 T + p T^{2} \) |
| 79 | \( 1 + 10 T + p T^{2} \) |
| 83 | \( 1 - 12 T + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 2 T + p T^{2} \) |
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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
−8.758223291095031754552956369438, −8.312260772544424185516721422091, −7.17873112796456375366578975968, −6.66551250524461407132951170236, −5.93258685270649161637474819784, −4.84958781245093527219072144847, −4.01099971240797084129426408389, −3.27066314296862874524604978075, −2.18987340870256474433797324505, −0.73714521852377401721058880406,
0.73714521852377401721058880406, 2.18987340870256474433797324505, 3.27066314296862874524604978075, 4.01099971240797084129426408389, 4.84958781245093527219072144847, 5.93258685270649161637474819784, 6.66551250524461407132951170236, 7.17873112796456375366578975968, 8.312260772544424185516721422091, 8.758223291095031754552956369438
