| L(s) = 1 | − 3-s − 3·7-s + 9-s + 2·11-s + 2·13-s + 4·17-s + 19-s + 3·21-s + 2·23-s − 27-s − 3·29-s + 4·31-s − 2·33-s − 6·37-s − 2·39-s − 11·41-s − 4·43-s + 2·49-s − 4·51-s + 3·53-s − 57-s + 3·59-s + 5·61-s − 3·63-s − 8·67-s − 2·69-s − 13·71-s + ⋯ |
| L(s) = 1 | − 0.577·3-s − 1.13·7-s + 1/3·9-s + 0.603·11-s + 0.554·13-s + 0.970·17-s + 0.229·19-s + 0.654·21-s + 0.417·23-s − 0.192·27-s − 0.557·29-s + 0.718·31-s − 0.348·33-s − 0.986·37-s − 0.320·39-s − 1.71·41-s − 0.609·43-s + 2/7·49-s − 0.560·51-s + 0.412·53-s − 0.132·57-s + 0.390·59-s + 0.640·61-s − 0.377·63-s − 0.977·67-s − 0.240·69-s − 1.54·71-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & \, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 91200 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & \, \Lambda(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.162575763\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.162575763\) |
| \(L(\frac{3}{2})\) |
|
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 | \( 1 \) |
| 19 | \( 1 - T \) |
| good | 7 | \( 1 + 3 T + p T^{2} \) |
| 11 | \( 1 - 2 T + p T^{2} \) |
| 13 | \( 1 - 2 T + p T^{2} \) |
| 17 | \( 1 - 4 T + p T^{2} \) |
| 23 | \( 1 - 2 T + p T^{2} \) |
| 29 | \( 1 + 3 T + p T^{2} \) |
| 31 | \( 1 - 4 T + p T^{2} \) |
| 37 | \( 1 + 6 T + p T^{2} \) |
| 41 | \( 1 + 11 T + p T^{2} \) |
| 43 | \( 1 + 4 T + p T^{2} \) |
| 47 | \( 1 + p T^{2} \) |
| 53 | \( 1 - 3 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 - 5 T + p T^{2} \) |
| 67 | \( 1 + 8 T + p T^{2} \) |
| 71 | \( 1 + 13 T + p T^{2} \) |
| 73 | \( 1 + 15 T + p T^{2} \) |
| 79 | \( 1 - 2 T + p T^{2} \) |
| 83 | \( 1 + p T^{2} \) |
| 89 | \( 1 + 11 T + p T^{2} \) |
| 97 | \( 1 + 8 T + p T^{2} \) |
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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
−13.73687477504192, −13.15578102890472, −13.07630672141012, −12.20888302182566, −11.89389299780956, −11.60426539003154, −10.81595542067903, −10.32672677790550, −9.935532824536377, −9.520545982010116, −8.804320522684498, −8.502019000644283, −7.691392393869239, −7.090720400183824, −6.696005988490302, −6.249352816053296, −5.587660130549453, −5.290559571376135, −4.428144176206463, −3.875609236916297, −3.225903235390892, −2.945727834013100, −1.748396896501746, −1.270038521939381, −0.3770583490544366,
0.3770583490544366, 1.270038521939381, 1.748396896501746, 2.945727834013100, 3.225903235390892, 3.875609236916297, 4.428144176206463, 5.290559571376135, 5.587660130549453, 6.249352816053296, 6.696005988490302, 7.090720400183824, 7.691392393869239, 8.502019000644283, 8.804320522684498, 9.520545982010116, 9.935532824536377, 10.32672677790550, 10.81595542067903, 11.60426539003154, 11.89389299780956, 12.20888302182566, 13.07630672141012, 13.15578102890472, 13.73687477504192
