| L(s) = 1 | − 3-s + 9-s − 2·11-s − 6·13-s − 2·17-s − 4·19-s + 4·23-s − 27-s − 5·31-s + 2·33-s − 7·37-s + 6·39-s + 10·41-s + 9·43-s − 6·47-s + 2·51-s − 14·53-s + 4·57-s + 10·59-s − 15·61-s + 4·67-s − 4·69-s + 2·71-s + 9·73-s − 79-s + 81-s + 10·83-s + ⋯ |
| L(s) = 1 | − 0.577·3-s + 1/3·9-s − 0.603·11-s − 1.66·13-s − 0.485·17-s − 0.917·19-s + 0.834·23-s − 0.192·27-s − 0.898·31-s + 0.348·33-s − 1.15·37-s + 0.960·39-s + 1.56·41-s + 1.37·43-s − 0.875·47-s + 0.280·51-s − 1.92·53-s + 0.529·57-s + 1.30·59-s − 1.92·61-s + 0.488·67-s − 0.481·69-s + 0.237·71-s + 1.05·73-s − 0.112·79-s + 1/9·81-s + 1.09·83-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 235200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(0.1553113487\) |
| \(L(\frac12)\) |
\(\approx\) |
\(0.1553113487\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 2 | \( 1 \) |
| 3 | \( 1 + T \) |
| 5 | \( 1 \) |
| 7 | \( 1 \) |
| good | 11 | \( 1 + 2 T + p T^{2} \) |
| 13 | \( 1 + 6 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 19 | \( 1 + 4 T + p T^{2} \) |
| 23 | \( 1 - 4 T + p T^{2} \) |
| 29 | \( 1 + p T^{2} \) |
| 31 | \( 1 + 5 T + p T^{2} \) |
| 37 | \( 1 + 7 T + p T^{2} \) |
| 41 | \( 1 - 10 T + p T^{2} \) |
| 43 | \( 1 - 9 T + p T^{2} \) |
| 47 | \( 1 + 6 T + p T^{2} \) |
| 53 | \( 1 + 14 T + p T^{2} \) |
| 59 | \( 1 - 10 T + p T^{2} \) |
| 61 | \( 1 + 15 T + p T^{2} \) |
| 67 | \( 1 - 4 T + p T^{2} \) |
| 71 | \( 1 - 2 T + p T^{2} \) |
| 73 | \( 1 - 9 T + p T^{2} \) |
| 79 | \( 1 + T + p T^{2} \) |
| 83 | \( 1 - 10 T + p T^{2} \) |
| 89 | \( 1 + 6 T + p T^{2} \) |
| 97 | \( 1 + 3 T + p T^{2} \) |
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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
−12.81111713778468, −12.50679297170883, −12.13475038451729, −11.39665616570286, −11.00551508081114, −10.70726517895746, −10.21633277044049, −9.594901201322590, −9.284010333960702, −8.798115352788989, −8.079949893479506, −7.576282008157336, −7.326232781369286, −6.608156072239153, −6.363775281220237, −5.558917521452693, −5.192494902305111, −4.732070724815297, −4.298998056734781, −3.623511474544046, −2.879096630499045, −2.382502876956852, −1.901065648742178, −1.055142971096556, −0.1201412387221429,
0.1201412387221429, 1.055142971096556, 1.901065648742178, 2.382502876956852, 2.879096630499045, 3.623511474544046, 4.298998056734781, 4.732070724815297, 5.192494902305111, 5.558917521452693, 6.363775281220237, 6.608156072239153, 7.326232781369286, 7.576282008157336, 8.079949893479506, 8.798115352788989, 9.284010333960702, 9.594901201322590, 10.21633277044049, 10.70726517895746, 11.00551508081114, 11.39665616570286, 12.13475038451729, 12.50679297170883, 12.81111713778468
